CalculusVolume2-SASG-05-02
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 5
Sequences and Series
5.2 Infinite Series
Section Exercises
Using sigma notation, write the following expressions as infinite series.
67.
Answer
: 69.
Answer
: Compute the first four partial sums for the series having term starting with as follows.
71.
Answer: 73.
Answer: In the following exercises, compute the general term of the series with the given partial sum If the sequence of partial sums converges, find its limit 75.
,
Answer: Series converges to
OpenStax Calculus Volume 2
Student Answer and Solution Guide
77.
Answer: Series diverges because partial sums are unbounded.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges.
79.
Answer:
In general Series diverges.
81.
(
Hint:
Use a partial fraction decomposition like that for Answer: The pattern is and the series converges to Suppose that that that and Find the sum of the indicated series.
83.
Answer: 85.
Answer: State whether the given series converges and explain why.
87.
(
Hint:
Rewrite using a change of index.)
Answer: diverges,
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
OpenStax Calculus Volume 2
Student Answer and Solution Guide
89.
Answer: convergent geometric series, 91.
1
+
π
e
2
+
π
2
e
4
+
π
3
e
6
+
π
4
e
8
+
⋯
Answer: convergent geometric series, For as follows, write the sum as a geometric series of the form State whether the
series converges and if it does, find the value of 93.
and for Answer: converges to 95.
and for Answer: converges to Use the identity to express the function as a geometric series in the indicated term.
97.
in Answer: 99.
in Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Evaluate the following telescoping series or state whether the series diverges.
101.
Answer: as 103.
Answer: diverges
Express the following series as a telescoping sum and evaluate its n
th partial sum.
105.
Answer: 107.
Answer:
and A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.
109.
Let in which as Find Answer: 111.
Suppose that where
as Find a condition on the coefficients that make this a general telescoping series.
Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
OpenStax Calculus Volume 2
Student Answer and Solution Guide
113.
Evaluate Answer: 115.
[T]
Define a sequence Use the graph of to verify that is increasing. Plot for and state whether it appears that the sequence converges.
Answer: converges to is a sum of rectangles of height over the interval
which lie above the graph of Each of the following infinite series converges to the given multiple of or In each case, find the minimum value of such that the partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to decimals place, 117.
[T]
error Answer:
119.
[T]
error Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
121.
[
T]
A fair coin is one that has probability of coming up heads when flipped.
a.
What is the probability that a fair coin will come up tails times in a row?
b.
Find the probability that a coin comes up heads for the first time on the last of an even number of coin flips.
Answer: a. The probability of any given ordered sequence of outcomes for coin flips is b. The probability of coming up heads for the first time on the th flip is the probability of the sequence which is The probability of coming up heads for the first time on an even flip is or 123.
[T]
Find the probability that a fair coin will come up heads for the second time after an even number of flips.
Answer: 125.
[T]
The expected number of times that a fair coin will come up heads is defined as the sum over of times the probability that the coin will come up heads exactly
times in a row, or Compute the expected number of consecutive times that a fair coin will come up heads. Answer:
as can be shown using summation by parts
127.
[
T]
Suppose that the amount of a drug in a patient’s system diminishes by a multiplicative factor each hour. Suppose that a new dose is administered every hours. Find an expression that gives the amount in the patient’s system after hours for each
in terms of the dosage and the ratio (Hint: Write where and sum over values from the different doses administered.)
Answer
: The part of the first dose after hours is the part of the second dose is and, in general, the part remaining of the dose is so
129.
Suppose that is a sequence of numbers. Explain why the sequence of partial sums of is increasing.
Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
131.
[T]
Suppose that and that, for given numbers and one defines
and Does converge? If so, to what? (First argue that
for all and is increasing.)
Answer: Since and since If for some n
,
then there is a smallest n
.
For this n
, so a contradiction. Thus and for all n
, so is increasing and bounded by Let If then but we can find n
such that which implies that contradicting that is increasing to Thus 133.
[T]
Suppose that is a convergent series of positive terms. Explain why
Answer: Let and Then eventually becomes arbitrarily close to which means that becomes arbitrarily small as 135.
[T]
Find the total length of the dashed path in the following figure.
Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
137.
[T]
The Sierpinski gasket is obtained by dividing the unit square into nine equal sub-
squares, removing the middle square, then doing the same at each stage to the remaining sub-squares. The figure shows the remaining set after four iterations. Compute the total area removed after stages, and compute the length the total perimeter of the remaining
set after stages.
Answer: At stage one a square of area is removed, at stage one removes squares of area
at stage three one removes squares of area and so on. The total removed area after stages is as The total perimeter is
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Student Project
Euler’s Constant
1.
Let Evaluate for various values of Answer: 3.
Now estimate how far is from
for a given integer Prove that for by using the following steps.
a.
Show that b.
Use the result from part a. to show that for any integer c.
For any integers and such that , express as a telescoping sum by writing
Use the result from part b. combined with this telescoping sum to conclude that
d.
Apply the limit to both sides of the inequality in part c. to conclude that
e.
Estimate
to an accuracy of within Answer:
a.
b.
Therefore, using the result from a. we conclude that
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
c.
d.
Therefore, e.
To estimate to an accuracy of within 0.001, we need to evaluate This file is copyright 2016, Rice University. All Rights Reserved.