CalculusVolume2-SASG-05-06
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 5
Sequences and Series
5.6 Ratio and Root Tests
Section Exercises
Use the ratio test to determine whether converges, where is given in the following problems. State if the ratio test is inconclusive.
317.
Answer: Converges.
319.
Answer: Converges.
321.
Answer: Converges.
323.
Answer: Converges.
325.
Answer: Ratio test is inconclusive.
327.
Answer: Converges.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Use the root test to determine whether converges, where is as follows.
329.
Answer: Diverges.
331.
Answer: Converges.
333.
Answer: Converges.
335.
Answer: Converges.
337.
Answer: by L’Hôpital’s rule. Converges.
In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series with given terms converges, or state if the test is inconclusive.
339.
Answer: Converges by ratio test.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
341.
Answer: Converges by root test.
343.
Answer: Diverges by root test.
Use the ratio test to determine whether converges, or state if the ratio test is inconclusive.
345.
Answer: Converge.
Use the root and limit comparison tests to determine whether converges
.
347.
where (
Hint:
Find limit of ) Answer: Converges by root test and limit comparison test since In the following exercises, use an appropriate test to determine whether the series converges.
349.
Answer: Converges absolutely by limit comparison with
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
351.
Answer: Series diverges.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
353.
Answer: Terms do not tend to zero: since 355.
where Answer: which converges by comparison with for 357.
Answer: converges by comparison with geometric series.
359.
(
Hint:
)
Answer: Series converges by limit comparison with
The following series converge by the ratio test. Use summation by parts,
to find the sum of the given series.
361.
where (
Hint:
Take and )
Answer: If and then and
363.
Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
The k
th term of each of the following series has a factor Find the range of for which the ratio test implies that the series converges.
365.
Answer: 367.
Answer: 369.
Let For which real numbers does converge?
Answer: All real numbers by the ratio test.
371.
Suppose that For which values of is guaranteed to converge?
Answer: 373.
For which values of if any, does converge? (
Hint:
Answer: Note that the ratio and root tests are inconclusive. Using the hint, there are
terms for and for each term is at least Thus,
which converges by the ratio test for For the series diverges by the divergence test.
375.
Let where is the greatest integer less than or equal to Determine whether converges and justify your answer.
Answer: One has The ratio test does not apply because if is even. However, so the series converges according to the previous exercise. Of course, the series is just a duplicated geometric series.
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
The following advanced
exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if then converges, while if then diverges.
377.
Let Show that For which does the generalized ratio test imply convergence of (
Hint:
Write
as a product of factors each smaller than )
Answer: The inverse of the factor is
so the product is less than Thus for
The series converges for
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter Review Exercises
True or False.
Justify your answer with a proof or a counterexample
.
379.
If then converges.
Answer: false
381.
If converges, then converges.
Answer: true
Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.
383.
Answer: unbounded, not monotone, divergent
385.
Answer: bounded, monotone, convergent, 387.
Answer: unbounded, not monotone, divergent
Is the series convergent or divergent?
389.
Answer: diverges
391.
Answer: converges
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Is the series convergent or divergent? If convergent, is it absolutely convergent?
393.
Answer: converges, but not absolutely
395.
Answer: converges absolutely
397.
Answer: converges absolutely
Evaluate
399.
Answer:
The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula where is the population of houseflies at generation and is the average number of offspring per housefly who survive to the next generation. Assume a starting population 401.
Find if and Answer:
403.
If and find and Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Student Project
Series Converging to and 1.
The series
was discovered by Gregory and Leibniz in the late This result follows from the Maclaurin series for We will discuss this series in the next chapter.
a.
Prove that this series converges.
b.
Evaluate the partial sums for c.
Use the remainder estimate for alternating series to get a bound on the error d.
What is the smallest value of that guarantees Evaluate Answer: a.
This series is an alternating series of the form where Since
we conclude that Therefore,
is a decreasing sequence. Further, Therefore, by the alternating series test, this series converges.
b.
c.
d.
We need Therefore, we need which implies .
3.
The series
was discovered by Ramanujan in the early
William Gosper, Jr., used this series to calculate to an accuracy of more than million digits in the At the time, that was a world record. Since that time, this series and others by Ramanujan have led mathematicians to find many other series representations for and a.
Prove that this series converges.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
b.
Evaluate the first term in this series. Compare this number with the value of from a calculating utility. To how many decimal places do these two numbers agree? What if we
add the first two terms in the series?
c.
Investigate the life of Srinivasa Ramanujan (1887 – 1920) and write a brief summary. Ramanujan is one of the most fascinating stories in the history of mathematics. He was basically self-taught, with no formal training in mathematics, yet he contributed in highly
original ways to many advanced areas of mathematics. Answer:
a.
Therefore, Therefore, by the ratio test, the series converges.
b.
The first term in this series is 0.31830987844. Using this value to approximate , we estimate by 3.14159273001. This approximation agrees with through the first six decimal places. The sum of the first two terms in the series is 0.31830988618. Using this value to approximate , we estimate by 3.14159265359.
c.
Answers will vary.
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