CalculusVolume2-SASG-05-03
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 5
Sequences and Series
5.3 The Divergence and Integral Tests
Section Exercises
For each of the following series, if the divergence test applies, either state that does not exist or find If the divergence test does not apply, state why.
139.
Answer:
Divergence test does not apply.
141.
Answer:
Series diverges.
143.
Answer:
(does not exist). Series diverges.
145.
Answer:
Series diverges.
147.
Answer:
does not exist. Series diverges.
149.
Answer:
Series diverges.
151.
Answer:
Divergence test does not apply.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
OpenStax Calculus Volume 2
Student Answer and Solution Guide
State whether the given converges.
153.
Answer: Series converges, 155.
Answer: Series converges, 157.
Answer: Series converges,
Use the integral test to determine whether the following sums converge.
159.
Answer: Series diverges by comparison with 161.
Answer: Series diverges by comparison with 163.
Answer: Series converges by comparison with Express the following sums as and determine whether each converges.
165.
(
Hint:
)
Answer: Since diverges by
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
167.
Answer: Since diverges by Use the estimate to find a bound for the remainder where
169.
Answer: 171.
Answer: [T] Find the minimum value of such that the remainder estimate guarantees that estimates accurate to within the given error.
173.
error Answer: 175.
error Answer: 177.
error
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: In the following exercises, find a value of such that is smaller than the desired error. Compute the corresponding sum and compare it to the given estimate of the infinite series.
179.
error Answer:
okay if Estimate agrees with to five decimal places.
181.
error Answer: okay if Estimate
agrees with the sum to four decimal places.
183.
Find the limit as of (
Hint:
Compare to )
Answer: The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise.
185.
In certain applications of probability, such as the so-called Watterson estimator for predicting mutation rates in population genetics, it is important to have an accurate estimate of the number
. Recall that is decreasing.
Compute to four decimal places. (
Hint:
)
Answer: 187.
[
T]
The simplest way to shuffle cards is to take the top card and insert it at a random place in the deck, called top random insertion, and then repeat. We will consider a deck to be randomly shuffled once enough top random insertions have been made that the card
OpenStax Calculus Volume 2
Student Answer and Solution Guide
originally at the bottom has reached the top and then been randomly inserted. If the deck has cards, then the probability that the insertion will be below the card initially at the bottom (call this card
) is Thus the expected number of top random insertions before is no longer at the bottom is n
. Once one card is below there are two places below and the probability that a randomly inserted card will fall below is
The expected number of top random insertions before this happens is The two cards below are now in random order. Continuing this way, find a formula for the expected number of top random insertions needed to consider the deck to be randomly shuffled.
Answer: The expected number of random insertions to get to the top is
Then one more insertion puts back in at random. Thus, the expected number of shuffles to randomize the deck is 189.
Show that for the remainder estimate to apply on it is sufficient that be decreasing on , but need not be decreasing on Answer: Set and such that is decreasing on 191.
Does converge if
is large enough? If so, for which Answer: The series converges for by integral test using change of variable.
193.
[T]
A fast computer can sum one million terms per second of the divergent series
Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed Answer: terms are needed.
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