CalculusVolume2-SASG-05-05
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 5
Sequences and Series
5.5 Alternating Series
Section Exercises
State whether each of the following series converges absolutely, conditionally, or not at all.
251.
Answer: Does not converge by divergence test. Terms do not tend to zero.
253.
Answer: Converges conditionally by alternating series test, since is decreasing. Does not converge absolutely by comparison with
p
-series, 255.
Answer: Converges absolutely by limit comparison to for example.
257.
Answer: Diverges by divergence test since 259.
Answer: Does not converge. Terms do not tend to zero.
261.
Answer: Diverges by divergence test.
263.
Answer: Converges by alternating series test.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
265.
Answer: Converges conditionally by alternating series test. Does not converge absolutely by limit comparison with
p
-series,
OpenStax Calculus Volume 2
Student Answer and Solution Guide
267.
Answer: Diverges; terms do not tend to zero.
269.
(
Hint:
for large )
Answer: Converges by alternating series test. Does not converge absolutely by limit comparison with harmonic series.
271.
(
Hint:
Find common denominator then rationalize numerator.)
Answer: Converges absolutely by limit comparison with
p
-series, after applying the hint.
273.
(
Hint:
Use Mean Value Theorem.) Answer: Converges by alternating series test since is decreasing to zero for large Does not converge absolutely by limit comparison with harmonic series after applying hint.
275.
Answer: Converges absolutely, since are terms of a telescoping series.
277.
Answer: Terms do not tend to zero. Series diverges by divergence test.
279.
Answer: Converges by alternating series test. Does not converge absolutely by limit comparison with harmonic series.
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
In each of the following problems, use the estimate to find a value of that guarantees that the sum of the first terms of the alternating series differs from the infinite sum by at most the given error. Calculate the partial sum for this
281.
[T]
error Answer:
283.
[T]
error Answer: or or 285.
[T]
error Answer: or For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.
287.
If is decreasing, then converges absolutely.
Answer: True. need not tend to zero since if then 289.
If is decreasing and converges then converges.
Answer: True. so convergence of follows from the comparison test.
291.
Let if and if (Also, and
If converges conditionally but not absolutely, then neither nor converge.
Answer: True. If one converges, then so must the other, implying absolute convergence.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
293.
Suppose that is a sequence such that converges for every possible sequence
of zeros and ones. Does converge absolutely?
Answer: Yes. Take if and if Then converges. Similarly, one can show converges. Since both series converge, the series must converge
absolutely.
The following series do not satisfy the hypotheses of the alternating series test as stated.
In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.
295.
Answer: Not decreasing. Does not converge absolutely.
297.
Answer: Not alternating. Can be expressed as which diverges by comparison with 299.
Suppose that converges absolutely. Show that the series consisting of the positive terms also converges.
Answer: Let if and if Then for all so the sequence of partial sums of is increasing and bounded above by the sequence of partial sums of which converges; hence, converges.
301.
The formula will be derived in the next chapter. Use the remainder to find a bound for the error in estimating by the fifth partial sum for and
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: For one has When When
When
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
303.
How many terms in are needed to approximate accurate to an error of at most Answer: Let Then when or and
whereas 305.
Sometimes the alternating series converges to a certain fraction of an absolutely convergent series at a faster rate. Given that find
Which of the series and gives a better estimation of using terms?
Answer: Let Then so The alternating series is more accurate for terms.
The following alternating series converge to given multiples of Find the value of predicted by the remainder estimate such that the partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum for which the error bound holds, and give the desired approximate value in each case. Up to decimals places,
307.
[T]
error Answer: 309.
[T]
If what is Answer:
The
partial sum is the same as that for the alternating harmonic series.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
311.
[
T]
Plot the series for and comment on its behavior
Answer: The series jumps rapidly near the endpoints. For away from the endpoints, the graph looks like 313.
[T]
The alternating harmonic series converges because of cancellation among its terms. Its sum is known because the cancellation can be described explicitly. A random harmonic series is one of the form where is a randomly generated sequence of
in which the values are equally likely to occur. Use a random number generator to produce random and plot the partial sums of your random harmonic sequence for to Compare to a plot of the first partial sums of the harmonic series. Answer: Here is a typical result. The top curve consists of partial sums of the harmonic series. The bottom curve plots partial sums of a random harmonic series.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
315.
[
T]
The Euler transform
rewrites as For the
alternating harmonic series, it takes the form Compute partial sums of until they approximate accurate to within How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate Answer: By the alternating series test, so one needs terms of the alternating harmonic series to estimate to within The first partial sums of the series
are (up to four decimals)
and the tenth partial sum is within of This file is copyright 2016, Rice University. All Rights Reserved.
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