Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The terms of the sequence { a n } increase in magnitude, so the limit of the sequence does not exist. b. The terms of the series ∑ 1 / k approach zero, so the series converges. c. The terms of the sequence of partial sums of the series approach so the infinite series ∑ a k approach 5 2 , so the infinite series converges to 5 2 . d. An alternating series that converges absolutely must converge conditionally. e. The sequence a n = n 2 n 2 + 1 converges. f. The sequence a n = ( − 1 ) n n 2 n 2 + 1 converges. g. The series ∑ k = 1 ∞ k 2 k 2 + 1 converges. h. The sequence of partial sums associated with the series ∑ k = 1 ∞ k 2 k 2 + 1 converges.

Question

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Answer 1

Textbook Question

Chapter 8, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The terms of the sequence {a_n} increase in magnitude, so the limit of the sequence does not exist.
b. The terms of the series $∑ 1 / k$ approach zero, so the series converges.
c. The terms of the sequence of partial sums of the series approach so the infinite series ∑a_k approach $5 2$ , so the infinite series converges to $5 2$ .
d. An alternating series that converges absolutely must converge conditionally.
e. The sequence $a n = n 2 n 2 + 1$ converges.
f. The sequence $a n = ( − 1 ) n n 2 n 2 + 1$ converges.
g. The series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.
h. The sequence of partial sums associated with the series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.

a.

Expert Solution

To determine

Whether the statement “The terms of the sequence ${an}$ increase in magnitude, so the limit of the sequence does not exist” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The n the term of the sequence is given as $an=2−1n2$ .

Obtain the limit of $an=2−1n2$ .

$limn→∞an=limn→∞(2−1n2)=limn→∞(2)−limn→∞(1n2)=2−1∞=2$

That is, the limit exists.

Hence, the given statement is false.

b.

Expert Solution

To determine

Whether the statement “The terms of the series $∑1k$ approach zero, so the series converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the series converges, then $limk→∞ak=0$ . Converse need not be true.

Calculation:

Consider k th term of the series as $ak=1k$ .

Obtain the limit of $ak$ .

$limk→∞ak=limk→∞1k=1∞=0$

Thus, by the result stated above note that the terms of the sequence tend to zero is not sufficient for convergence of the series.

Hence, the given statement is false.

c.

Expert Solution

To determine

Whether the statement “The terms of the sequence of partial sums of the series $∑ak$ approach $52$ , so the infinite series converges to $52$ ” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Definition used:

If the sequence of partial sums ${Sn}$ has a limit $L$ , the infinite series converges to that limit.

Calculation:

From the given statement, the terms of the sequence of partial sums of the series $∑ak$ approach $52$ .

By the definition stated above, the infinite series converges to $52$ .

Hence, the given statement is true.

d.

Expert Solution

To determine

Whether the statement “An alternating series that converges absolutely must converge conditionally” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Definition used:

(1) The series $∑ak$ converges absolutely if $∑ak and ∑|ak|$ converges.

(2) The series $∑ak$ converges conditionally if $∑ak$ converges but $∑|ak|$ diverges.

Calculation:

From the given statement, the series converges absolutely.

By the definition (1) stated above, the series $∑ak and ∑|ak|$ both are converges.

Since the absolute value of the $∑|ak|$ converges and by the definition (2) stated above, it can be concluded that the series does not conditionally converge.

Hence, the given statement is false.

e.

Expert Solution

To determine

Whether the statement “The sequence $an=n2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞n2n2+1=limn→∞n2n2(1+1n2)=1(1+1∞)=1$

Since the limit of the sequence exist and result stated above, it can be concluded that the sequence converges.

Hence, the given statement is true.

f.

Expert Solution

To determine

Whether the statement “The sequence $an=(−1)nn2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges otherwise it is diverges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞(−1)nn2n2+1=limn→∞(−1)nn2n2(1+1n2)=(−1)n(1+1∞)=(−1)n$

If $n$ is odd, the sequence has limit $−1$ and if n is even the sequence has limit 1.

That is, the sequence does not have the same limit.

Thus, by result stated above, it can be concluded that the sequence does not converge.

Hence, the given statement is false.

g.

Expert Solution

To determine

Whether the statement “The series $∑k=1∞k2k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If $limk→∞ak≠0$ , then the series diverges.

Calculation:

Obtain the limit of $ak=k2k2+1$ .

$limk→∞ak=limk→∞k2k2+1=limk→∞k2k2(1+1k2)=1(1+1∞)=1$

Since $limk→∞ak≠0$ and by result stated above, it can be concluded that the series diverges.

Hence, the given statement is false.

h.

Expert Solution

To determine

Whether the statement “The sequence of partial sums association with the series $∑k=1∞1k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

(1) “Let $∑ak$ and $∑bk$ be series with positive terms.

(a) If $0<ak≤bk$ and $∑bk$ converges, then $∑ak$ converges.
(b) If $0<bk≤ak$ and $∑bk$ diverges, then $∑ak$ diverges.

(2) The p-series $∑k=1∞1kp$ is convergent if $p>1$ and diverges if $p≤1$ .

Calculation:

Let $k2<k2+1$

$1k2+1<1k2∑k=1∞1k2+1<∑k=1∞1k2$

In the series $∑k=1∞1k2$ , $p=2$ . That is, the value of p is greater than 1.

Thus, by the Result (2), the series $∑k=1∞1k2$ is convergent.

Therefore, by the Result (1), it can be concluded that the given series convergent.

Hence, the series $∑k=1∞1k2+1$ converges.

Hence, the given statement is true.

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Answer 2

Textbook Question

Chapter 8, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The terms of the sequence {a_n} increase in magnitude, so the limit of the sequence does not exist.
b. The terms of the series $∑ 1 / k$ approach zero, so the series converges.
c. The terms of the sequence of partial sums of the series approach so the infinite series ∑a_k approach $5 2$ , so the infinite series converges to $5 2$ .
d. An alternating series that converges absolutely must converge conditionally.
e. The sequence $a n = n 2 n 2 + 1$ converges.
f. The sequence $a n = ( − 1 ) n n 2 n 2 + 1$ converges.
g. The series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.
h. The sequence of partial sums associated with the series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.

a.

Expert Solution

To determine

Whether the statement “The terms of the sequence ${an}$ increase in magnitude, so the limit of the sequence does not exist” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The n the term of the sequence is given as $an=2−1n2$ .

Obtain the limit of $an=2−1n2$ .

$limn→∞an=limn→∞(2−1n2)=limn→∞(2)−limn→∞(1n2)=2−1∞=2$

That is, the limit exists.

Hence, the given statement is false.

b.

Expert Solution

To determine

Whether the statement “The terms of the series $∑1k$ approach zero, so the series converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the series converges, then $limk→∞ak=0$ . Converse need not be true.

Calculation:

Consider k th term of the series as $ak=1k$ .

Obtain the limit of $ak$ .

$limk→∞ak=limk→∞1k=1∞=0$

Thus, by the result stated above note that the terms of the sequence tend to zero is not sufficient for convergence of the series.

Hence, the given statement is false.

c.

Expert Solution

To determine

Whether the statement “The terms of the sequence of partial sums of the series $∑ak$ approach $52$ , so the infinite series converges to $52$ ” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Definition used:

If the sequence of partial sums ${Sn}$ has a limit $L$ , the infinite series converges to that limit.

Calculation:

From the given statement, the terms of the sequence of partial sums of the series $∑ak$ approach $52$ .

By the definition stated above, the infinite series converges to $52$ .

Hence, the given statement is true.

d.

Expert Solution

To determine

Whether the statement “An alternating series that converges absolutely must converge conditionally” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Definition used:

(1) The series $∑ak$ converges absolutely if $∑ak and ∑|ak|$ converges.

(2) The series $∑ak$ converges conditionally if $∑ak$ converges but $∑|ak|$ diverges.

Calculation:

From the given statement, the series converges absolutely.

By the definition (1) stated above, the series $∑ak and ∑|ak|$ both are converges.

Since the absolute value of the $∑|ak|$ converges and by the definition (2) stated above, it can be concluded that the series does not conditionally converge.

Hence, the given statement is false.

e.

Expert Solution

To determine

Whether the statement “The sequence $an=n2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞n2n2+1=limn→∞n2n2(1+1n2)=1(1+1∞)=1$

Since the limit of the sequence exist and result stated above, it can be concluded that the sequence converges.

Hence, the given statement is true.

f.

Expert Solution

To determine

Whether the statement “The sequence $an=(−1)nn2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges otherwise it is diverges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞(−1)nn2n2+1=limn→∞(−1)nn2n2(1+1n2)=(−1)n(1+1∞)=(−1)n$

If $n$ is odd, the sequence has limit $−1$ and if n is even the sequence has limit 1.

That is, the sequence does not have the same limit.

Thus, by result stated above, it can be concluded that the sequence does not converge.

Hence, the given statement is false.

g.

Expert Solution

To determine

Whether the statement “The series $∑k=1∞k2k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If $limk→∞ak≠0$ , then the series diverges.

Calculation:

Obtain the limit of $ak=k2k2+1$ .

$limk→∞ak=limk→∞k2k2+1=limk→∞k2k2(1+1k2)=1(1+1∞)=1$

Since $limk→∞ak≠0$ and by result stated above, it can be concluded that the series diverges.

Hence, the given statement is false.

h.

Expert Solution

To determine

Whether the statement “The sequence of partial sums association with the series $∑k=1∞1k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

(1) “Let $∑ak$ and $∑bk$ be series with positive terms.

(a) If $0<ak≤bk$ and $∑bk$ converges, then $∑ak$ converges.
(b) If $0<bk≤ak$ and $∑bk$ diverges, then $∑ak$ diverges.

(2) The p-series $∑k=1∞1kp$ is convergent if $p>1$ and diverges if $p≤1$ .

Calculation:

Let $k2<k2+1$

$1k2+1<1k2∑k=1∞1k2+1<∑k=1∞1k2$

In the series $∑k=1∞1k2$ , $p=2$ . That is, the value of p is greater than 1.

Thus, by the Result (2), the series $∑k=1∞1k2$ is convergent.

Therefore, by the Result (1), it can be concluded that the given series convergent.

Hence, the series $∑k=1∞1k2+1$ converges.

Hence, the given statement is true.

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Answer 3

Textbook Question

Chapter 8, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The terms of the sequence {a_n} increase in magnitude, so the limit of the sequence does not exist.
b. The terms of the series $∑ 1 / k$ approach zero, so the series converges.
c. The terms of the sequence of partial sums of the series approach so the infinite series ∑a_k approach $5 2$ , so the infinite series converges to $5 2$ .
d. An alternating series that converges absolutely must converge conditionally.
e. The sequence $a n = n 2 n 2 + 1$ converges.
f. The sequence $a n = ( − 1 ) n n 2 n 2 + 1$ converges.
g. The series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.
h. The sequence of partial sums associated with the series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.

a.

Expert Solution

To determine

Whether the statement “The terms of the sequence ${an}$ increase in magnitude, so the limit of the sequence does not exist” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The n the term of the sequence is given as $an=2−1n2$ .

Obtain the limit of $an=2−1n2$ .

$limn→∞an=limn→∞(2−1n2)=limn→∞(2)−limn→∞(1n2)=2−1∞=2$

That is, the limit exists.

Hence, the given statement is false.

b.

Expert Solution

To determine

Whether the statement “The terms of the series $∑1k$ approach zero, so the series converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the series converges, then $limk→∞ak=0$ . Converse need not be true.

Calculation:

Consider k th term of the series as $ak=1k$ .

Obtain the limit of $ak$ .

$limk→∞ak=limk→∞1k=1∞=0$

Thus, by the result stated above note that the terms of the sequence tend to zero is not sufficient for convergence of the series.

Hence, the given statement is false.

c.

Expert Solution

To determine

Whether the statement “The terms of the sequence of partial sums of the series $∑ak$ approach $52$ , so the infinite series converges to $52$ ” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Definition used:

If the sequence of partial sums ${Sn}$ has a limit $L$ , the infinite series converges to that limit.

Calculation:

From the given statement, the terms of the sequence of partial sums of the series $∑ak$ approach $52$ .

By the definition stated above, the infinite series converges to $52$ .

Hence, the given statement is true.

d.

Expert Solution

To determine

Whether the statement “An alternating series that converges absolutely must converge conditionally” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Definition used:

(1) The series $∑ak$ converges absolutely if $∑ak and ∑|ak|$ converges.

(2) The series $∑ak$ converges conditionally if $∑ak$ converges but $∑|ak|$ diverges.

Calculation:

From the given statement, the series converges absolutely.

By the definition (1) stated above, the series $∑ak and ∑|ak|$ both are converges.

Since the absolute value of the $∑|ak|$ converges and by the definition (2) stated above, it can be concluded that the series does not conditionally converge.

Hence, the given statement is false.

e.

Expert Solution

To determine

Whether the statement “The sequence $an=n2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞n2n2+1=limn→∞n2n2(1+1n2)=1(1+1∞)=1$

Since the limit of the sequence exist and result stated above, it can be concluded that the sequence converges.

Hence, the given statement is true.

f.

Expert Solution

To determine

Whether the statement “The sequence $an=(−1)nn2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges otherwise it is diverges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞(−1)nn2n2+1=limn→∞(−1)nn2n2(1+1n2)=(−1)n(1+1∞)=(−1)n$

If $n$ is odd, the sequence has limit $−1$ and if n is even the sequence has limit 1.

That is, the sequence does not have the same limit.

Thus, by result stated above, it can be concluded that the sequence does not converge.

Hence, the given statement is false.

g.

Expert Solution

To determine

Whether the statement “The series $∑k=1∞k2k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If $limk→∞ak≠0$ , then the series diverges.

Calculation:

Obtain the limit of $ak=k2k2+1$ .

$limk→∞ak=limk→∞k2k2+1=limk→∞k2k2(1+1k2)=1(1+1∞)=1$

Since $limk→∞ak≠0$ and by result stated above, it can be concluded that the series diverges.

Hence, the given statement is false.

h.

Expert Solution

To determine

Whether the statement “The sequence of partial sums association with the series $∑k=1∞1k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

(1) “Let $∑ak$ and $∑bk$ be series with positive terms.

(a) If $0<ak≤bk$ and $∑bk$ converges, then $∑ak$ converges.
(b) If $0<bk≤ak$ and $∑bk$ diverges, then $∑ak$ diverges.

(2) The p-series $∑k=1∞1kp$ is convergent if $p>1$ and diverges if $p≤1$ .

Calculation:

Let $k2<k2+1$

$1k2+1<1k2∑k=1∞1k2+1<∑k=1∞1k2$

In the series $∑k=1∞1k2$ , $p=2$ . That is, the value of p is greater than 1.

Thus, by the Result (2), the series $∑k=1∞1k2$ is convergent.

Therefore, by the Result (1), it can be concluded that the given series convergent.

Hence, the series $∑k=1∞1k2+1$ converges.

Hence, the given statement is true.

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Answer 4

Textbook Question

Chapter 8, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The terms of the sequence {a_n} increase in magnitude, so the limit of the sequence does not exist.
b. The terms of the series $∑ 1 / k$ approach zero, so the series converges.
c. The terms of the sequence of partial sums of the series approach so the infinite series ∑a_k approach $5 2$ , so the infinite series converges to $5 2$ .
d. An alternating series that converges absolutely must converge conditionally.
e. The sequence $a n = n 2 n 2 + 1$ converges.
f. The sequence $a n = ( − 1 ) n n 2 n 2 + 1$ converges.
g. The series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.
h. The sequence of partial sums associated with the series $∑ k = 1 ∞ k 2 k 2 + 1$ converges.

a.

Expert Solution

To determine

Whether the statement “The terms of the sequence ${an}$ increase in magnitude, so the limit of the sequence does not exist” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The n the term of the sequence is given as $an=2−1n2$ .

Obtain the limit of $an=2−1n2$ .

$limn→∞an=limn→∞(2−1n2)=limn→∞(2)−limn→∞(1n2)=2−1∞=2$

That is, the limit exists.

Hence, the given statement is false.

b.

Expert Solution

To determine

Whether the statement “The terms of the series $∑1k$ approach zero, so the series converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the series converges, then $limk→∞ak=0$ . Converse need not be true.

Calculation:

Consider k th term of the series as $ak=1k$ .

Obtain the limit of $ak$ .

$limk→∞ak=limk→∞1k=1∞=0$

Thus, by the result stated above note that the terms of the sequence tend to zero is not sufficient for convergence of the series.

Hence, the given statement is false.

c.

Expert Solution

To determine

Whether the statement “The terms of the sequence of partial sums of the series $∑ak$ approach $52$ , so the infinite series converges to $52$ ” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Definition used:

If the sequence of partial sums ${Sn}$ has a limit $L$ , the infinite series converges to that limit.

Calculation:

From the given statement, the terms of the sequence of partial sums of the series $∑ak$ approach $52$ .

By the definition stated above, the infinite series converges to $52$ .

Hence, the given statement is true.

d.

Expert Solution

To determine

Whether the statement “An alternating series that converges absolutely must converge conditionally” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Definition used:

(1) The series $∑ak$ converges absolutely if $∑ak and ∑|ak|$ converges.

(2) The series $∑ak$ converges conditionally if $∑ak$ converges but $∑|ak|$ diverges.

Calculation:

From the given statement, the series converges absolutely.

By the definition (1) stated above, the series $∑ak and ∑|ak|$ both are converges.

Since the absolute value of the $∑|ak|$ converges and by the definition (2) stated above, it can be concluded that the series does not conditionally converge.

Hence, the given statement is false.

e.

Expert Solution

To determine

Whether the statement “The sequence $an=n2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞n2n2+1=limn→∞n2n2(1+1n2)=1(1+1∞)=1$

Since the limit of the sequence exist and result stated above, it can be concluded that the sequence converges.

Hence, the given statement is true.

f.

Expert Solution

To determine

Whether the statement “The sequence $an=(−1)nn2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges otherwise it is diverges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞(−1)nn2n2+1=limn→∞(−1)nn2n2(1+1n2)=(−1)n(1+1∞)=(−1)n$

If $n$ is odd, the sequence has limit $−1$ and if n is even the sequence has limit 1.

That is, the sequence does not have the same limit.

Thus, by result stated above, it can be concluded that the sequence does not converge.

Hence, the given statement is false.

g.

Expert Solution

To determine

Whether the statement “The series $∑k=1∞k2k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If $limk→∞ak≠0$ , then the series diverges.

Calculation:

Obtain the limit of $ak=k2k2+1$ .

$limk→∞ak=limk→∞k2k2+1=limk→∞k2k2(1+1k2)=1(1+1∞)=1$

Since $limk→∞ak≠0$ and by result stated above, it can be concluded that the series diverges.

Hence, the given statement is false.

h.

Expert Solution

To determine

Whether the statement “The sequence of partial sums association with the series $∑k=1∞1k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

(1) “Let $∑ak$ and $∑bk$ be series with positive terms.

(a) If $0<ak≤bk$ and $∑bk$ converges, then $∑ak$ converges.
(b) If $0<bk≤ak$ and $∑bk$ diverges, then $∑ak$ diverges.

(2) The p-series $∑k=1∞1kp$ is convergent if $p>1$ and diverges if $p≤1$ .

Calculation:

Let $k2<k2+1$

$1k2+1<1k2∑k=1∞1k2+1<∑k=1∞1k2$

In the series $∑k=1∞1k2$ , $p=2$ . That is, the value of p is greater than 1.

Thus, by the Result (2), the series $∑k=1∞1k2$ is convergent.

Therefore, by the Result (1), it can be concluded that the given series convergent.

Hence, the series $∑k=1∞1k2+1$ converges.

Hence, the given statement is true.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

a.

Expert Solution

To determine

Whether the statement “The terms of the sequence ${an}$ increase in magnitude, so the limit of the sequence does not exist” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The n the term of the sequence is given as $an=2−1n2$ .

Obtain the limit of $an=2−1n2$ .

$limn→∞an=limn→∞(2−1n2)=limn→∞(2)−limn→∞(1n2)=2−1∞=2$

That is, the limit exists.

Hence, the given statement is false.

b.

Expert Solution

To determine

Whether the statement “The terms of the series $∑1k$ approach zero, so the series converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the series converges, then $limk→∞ak=0$ . Converse need not be true.

Calculation:

Consider k th term of the series as $ak=1k$ .

Obtain the limit of $ak$ .

$limk→∞ak=limk→∞1k=1∞=0$

Thus, by the result stated above note that the terms of the sequence tend to zero is not sufficient for convergence of the series.

Hence, the given statement is false.

c.

Expert Solution

To determine

Whether the statement “The terms of the sequence of partial sums of the series $∑ak$ approach $52$ , so the infinite series converges to $52$ ” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Definition used:

If the sequence of partial sums ${Sn}$ has a limit $L$ , the infinite series converges to that limit.

Calculation:

From the given statement, the terms of the sequence of partial sums of the series $∑ak$ approach $52$ .

By the definition stated above, the infinite series converges to $52$ .

Hence, the given statement is true.

d.

Expert Solution

To determine

Whether the statement “An alternating series that converges absolutely must converge conditionally” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Definition used:

(1) The series $∑ak$ converges absolutely if $∑ak and ∑|ak|$ converges.

(2) The series $∑ak$ converges conditionally if $∑ak$ converges but $∑|ak|$ diverges.

Calculation:

From the given statement, the series converges absolutely.

By the definition (1) stated above, the series $∑ak and ∑|ak|$ both are converges.

Since the absolute value of the $∑|ak|$ converges and by the definition (2) stated above, it can be concluded that the series does not conditionally converge.

Hence, the given statement is false.

e.

Expert Solution

To determine

Whether the statement “The sequence $an=n2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞n2n2+1=limn→∞n2n2(1+1n2)=1(1+1∞)=1$

Since the limit of the sequence exist and result stated above, it can be concluded that the sequence converges.

Hence, the given statement is true.

f.

Expert Solution

To determine

Whether the statement “The sequence $an=(−1)nn2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges otherwise it is diverges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞(−1)nn2n2+1=limn→∞(−1)nn2n2(1+1n2)=(−1)n(1+1∞)=(−1)n$

If $n$ is odd, the sequence has limit $−1$ and if n is even the sequence has limit 1.

That is, the sequence does not have the same limit.

Thus, by result stated above, it can be concluded that the sequence does not converge.

Hence, the given statement is false.

g.

Expert Solution

To determine

Whether the statement “The series $∑k=1∞k2k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If $limk→∞ak≠0$ , then the series diverges.

Calculation:

Obtain the limit of $ak=k2k2+1$ .

$limk→∞ak=limk→∞k2k2+1=limk→∞k2k2(1+1k2)=1(1+1∞)=1$

Since $limk→∞ak≠0$ and by result stated above, it can be concluded that the series diverges.

Hence, the given statement is false.

h.

Expert Solution

To determine

Whether the statement “The sequence of partial sums association with the series $∑k=1∞1k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

(1) “Let $∑ak$ and $∑bk$ be series with positive terms.

(a) If $0<ak≤bk$ and $∑bk$ converges, then $∑ak$ converges.
(b) If $0<bk≤ak$ and $∑bk$ diverges, then $∑ak$ diverges.

(2) The p-series $∑k=1∞1kp$ is convergent if $p>1$ and diverges if $p≤1$ .

Calculation:

Let $k2<k2+1$

$1k2+1<1k2∑k=1∞1k2+1<∑k=1∞1k2$

In the series $∑k=1∞1k2$ , $p=2$ . That is, the value of p is greater than 1.

Thus, by the Result (2), the series $∑k=1∞1k2$ is convergent.

Therefore, by the Result (1), it can be concluded that the given series convergent.

Hence, the series $∑k=1∞1k2+1$ converges.

Hence, the given statement is true.

Answer 8

a.

Expert Solution

To determine

Whether the statement “The terms of the sequence ${an}$ increase in magnitude, so the limit of the sequence does not exist” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The n the term of the sequence is given as $an=2−1n2$ .

Obtain the limit of $an=2−1n2$ .

$limn→∞an=limn→∞(2−1n2)=limn→∞(2)−limn→∞(1n2)=2−1∞=2$

That is, the limit exists.

Hence, the given statement is false.

Answer 9

a.

Expert Solution

Answer 10

a.

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

Whether the statement “The terms of the sequence ${an}$ increase in magnitude, so the limit of the sequence does not exist” is true or not.

Answer 13

Answer to Problem 1RE

The statement is false.

Answer 14

Explanation of Solution

The n the term of the sequence is given as $an=2−1n2$ .

Obtain the limit of $an=2−1n2$ .

$limn→∞an=limn→∞(2−1n2)=limn→∞(2)−limn→∞(1n2)=2−1∞=2$

That is, the limit exists.

Hence, the given statement is false.

Answer 15

b.

Expert Solution

To determine

Whether the statement “The terms of the series $∑1k$ approach zero, so the series converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the series converges, then $limk→∞ak=0$ . Converse need not be true.

Calculation:

Consider k th term of the series as $ak=1k$ .

Obtain the limit of $ak$ .

$limk→∞ak=limk→∞1k=1∞=0$

Thus, by the result stated above note that the terms of the sequence tend to zero is not sufficient for convergence of the series.

Hence, the given statement is false.

Answer 16

b.

Expert Solution

Answer 17

b.

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

Whether the statement “The terms of the series $∑1k$ approach zero, so the series converges” is true or not.

Answer 20

Answer to Problem 1RE

The statement is false.

Answer 21

Explanation of Solution

Result used:

If the series converges, then $limk→∞ak=0$ . Converse need not be true.

Calculation:

Consider k th term of the series as $ak=1k$ .

Obtain the limit of $ak$ .

$limk→∞ak=limk→∞1k=1∞=0$

Thus, by the result stated above note that the terms of the sequence tend to zero is not sufficient for convergence of the series.

Hence, the given statement is false.

Answer 22

c.

Expert Solution

To determine

Whether the statement “The terms of the sequence of partial sums of the series $∑ak$ approach $52$ , so the infinite series converges to $52$ ” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Definition used:

If the sequence of partial sums ${Sn}$ has a limit $L$ , the infinite series converges to that limit.

Calculation:

From the given statement, the terms of the sequence of partial sums of the series $∑ak$ approach $52$ .

By the definition stated above, the infinite series converges to $52$ .

Hence, the given statement is true.

Answer 23

c.

Expert Solution

Answer 24

c.

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

Whether the statement “The terms of the sequence of partial sums of the series $∑ak$ approach $52$ , so the infinite series converges to $52$ ” is true or not.

Answer 27

Answer to Problem 1RE

The statement is true.

Answer 28

Explanation of Solution

Definition used:

If the sequence of partial sums ${Sn}$ has a limit $L$ , the infinite series converges to that limit.

Calculation:

From the given statement, the terms of the sequence of partial sums of the series $∑ak$ approach $52$ .

By the definition stated above, the infinite series converges to $52$ .

Hence, the given statement is true.

Answer 29

d.

Expert Solution

To determine

Whether the statement “An alternating series that converges absolutely must converge conditionally” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Definition used:

(1) The series $∑ak$ converges absolutely if $∑ak and ∑|ak|$ converges.

(2) The series $∑ak$ converges conditionally if $∑ak$ converges but $∑|ak|$ diverges.

Calculation:

From the given statement, the series converges absolutely.

By the definition (1) stated above, the series $∑ak and ∑|ak|$ both are converges.

Since the absolute value of the $∑|ak|$ converges and by the definition (2) stated above, it can be concluded that the series does not conditionally converge.

Hence, the given statement is false.

Answer 30

d.

Expert Solution

Answer 31

d.

Expert Solution

Answer 32

Expert Solution

Answer 33

To determine

Whether the statement “An alternating series that converges absolutely must converge conditionally” is true or not.

Answer 34

Answer to Problem 1RE

The statement is false.

Answer 35

Explanation of Solution

Definition used:

(1) The series $∑ak$ converges absolutely if $∑ak and ∑|ak|$ converges.

(2) The series $∑ak$ converges conditionally if $∑ak$ converges but $∑|ak|$ diverges.

Calculation:

From the given statement, the series converges absolutely.

By the definition (1) stated above, the series $∑ak and ∑|ak|$ both are converges.

Since the absolute value of the $∑|ak|$ converges and by the definition (2) stated above, it can be concluded that the series does not conditionally converge.

Hence, the given statement is false.

Answer 36

e.

Expert Solution

To determine

Whether the statement “The sequence $an=n2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞n2n2+1=limn→∞n2n2(1+1n2)=1(1+1∞)=1$

Since the limit of the sequence exist and result stated above, it can be concluded that the sequence converges.

Hence, the given statement is true.

Answer 37

e.

Expert Solution

Answer 38

e.

Expert Solution

Answer 39

Expert Solution

Answer 40

To determine

Whether the statement “The sequence $an=n2n2+1$ converges” is true or not.

Answer 41

Answer to Problem 1RE

The statement is true.

Answer 42

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞n2n2+1=limn→∞n2n2(1+1n2)=1(1+1∞)=1$

Since the limit of the sequence exist and result stated above, it can be concluded that the sequence converges.

Hence, the given statement is true.

Answer 43

f.

Expert Solution

To determine

Whether the statement “The sequence $an=(−1)nn2n2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges otherwise it is diverges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞(−1)nn2n2+1=limn→∞(−1)nn2n2(1+1n2)=(−1)n(1+1∞)=(−1)n$

If $n$ is odd, the sequence has limit $−1$ and if n is even the sequence has limit 1.

That is, the sequence does not have the same limit.

Thus, by result stated above, it can be concluded that the sequence does not converge.

Hence, the given statement is false.

Answer 44

f.

Expert Solution

Answer 45

f.

Expert Solution

Answer 46

Expert Solution

Answer 47

To determine

Whether the statement “The sequence $an=(−1)nn2n2+1$ converges” is true or not.

Answer 48

Answer to Problem 1RE

The statement is false.

Answer 49

Explanation of Solution

Result used:

If the sequence has the limit, then the sequence converges otherwise it is diverges.

Calculation:

Obtain the limit of $an$ .

$limn→∞an=limn→∞(−1)nn2n2+1=limn→∞(−1)nn2n2(1+1n2)=(−1)n(1+1∞)=(−1)n$

If $n$ is odd, the sequence has limit $−1$ and if n is even the sequence has limit 1.

That is, the sequence does not have the same limit.

Thus, by result stated above, it can be concluded that the sequence does not converge.

Hence, the given statement is false.

Answer 50

g.

Expert Solution

To determine

Whether the statement “The series $∑k=1∞k2k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

Result used:

If $limk→∞ak≠0$ , then the series diverges.

Calculation:

Obtain the limit of $ak=k2k2+1$ .

$limk→∞ak=limk→∞k2k2+1=limk→∞k2k2(1+1k2)=1(1+1∞)=1$

Since $limk→∞ak≠0$ and by result stated above, it can be concluded that the series diverges.

Hence, the given statement is false.

Answer 51

g.

Expert Solution

Answer 52

g.

Expert Solution

Answer 53

Expert Solution

Answer 54

To determine

Whether the statement “The series $∑k=1∞k2k2+1$ converges” is true or not.

Answer 55

Answer to Problem 1RE

The statement is false.

Answer 56

Explanation of Solution

Result used:

If $limk→∞ak≠0$ , then the series diverges.

Calculation:

Obtain the limit of $ak=k2k2+1$ .

$limk→∞ak=limk→∞k2k2+1=limk→∞k2k2(1+1k2)=1(1+1∞)=1$

Since $limk→∞ak≠0$ and by result stated above, it can be concluded that the series diverges.

Hence, the given statement is false.

Answer 57

h.

Expert Solution

To determine

Whether the statement “The sequence of partial sums association with the series $∑k=1∞1k2+1$ converges” is true or not.

Answer to Problem 1RE

The statement is true.

Explanation of Solution

Result used:

(1) “Let $∑ak$ and $∑bk$ be series with positive terms.

(a) If $0<ak≤bk$ and $∑bk$ converges, then $∑ak$ converges.
(b) If $0<bk≤ak$ and $∑bk$ diverges, then $∑ak$ diverges.

(2) The p-series $∑k=1∞1kp$ is convergent if $p>1$ and diverges if $p≤1$ .

Calculation:

Let $k2<k2+1$

$1k2+1<1k2∑k=1∞1k2+1<∑k=1∞1k2$

In the series $∑k=1∞1k2$ , $p=2$ . That is, the value of p is greater than 1.

Thus, by the Result (2), the series $∑k=1∞1k2$ is convergent.

Therefore, by the Result (1), it can be concluded that the given series convergent.

Hence, the series $∑k=1∞1k2+1$ converges.

Hence, the given statement is true.

Answer 58

h.

Expert Solution

Answer 59

h.

Expert Solution

Answer 60

Expert Solution

Answer 61

To determine

Whether the statement “The sequence of partial sums association with the series $∑k=1∞1k2+1$ converges” is true or not.

Answer 62

Answer to Problem 1RE

The statement is true.

Answer 63

Explanation of Solution

Result used:

(1) “Let $∑ak$ and $∑bk$ be series with positive terms.

(a) If $0<ak≤bk$ and $∑bk$ converges, then $∑ak$ converges.
(b) If $0<bk≤ak$ and $∑bk$ diverges, then $∑ak$ diverges.

(2) The p-series $∑k=1∞1kp$ is convergent if $p>1$ and diverges if $p≤1$ .

Calculation:

Let $k2<k2+1$

$1k2+1<1k2∑k=1∞1k2+1<∑k=1∞1k2$

In the series $∑k=1∞1k2$ , $p=2$ . That is, the value of p is greater than 1.

Thus, by the Result (2), the series $∑k=1∞1k2$ is convergent.

Therefore, by the Result (1), it can be concluded that the given series convergent.

Hence, the series $∑k=1∞1k2+1$ converges.

Hence, the given statement is true.

Answer 64

Ch. 8.1 - Define sequence and give an example.Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Given the series k=1k, evaluate the first four...Ch. 8.1 - The terms of a sequence of partial sums are...Ch. 8.1 - Consider the infinite series k=11k. Evaluate the...Ch. 8.1 - Explicit formulas Write the first four terms of...Ch. 8.1 - Explicit formulas Write the first four terms of...

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