Stewart Section 11.6: Absolute Convergence, Ratio and Root Tests

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Stewart Section 11.6: Absolute Convergence, Ratio and Root Tests 0/53
1 Ratio Test - Concepts · Level 2
What can you say about the series \(\sum a_n\) in each of the following cases?
(a) \(\operatorname*{lim}\limits_{n \rightarrow \infty} |\dfrac{a_{n+1}}{a_n}| = 8\)
(b) \(\operatorname*{lim}\limits_{n \rightarrow \infty} |\dfrac{a_{n+1}}{a_n}| = 0.8\)
(c) \(\operatorname*{lim}\limits_{n \rightarrow \infty} |\dfrac{a_{n+1}}{a_n}| = 1\)

Enter your answer directly below each part above.

2 Absolute Convergence · Level 2
Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{\sqrt{n}}\)
3 Absolute Convergence · Level 2
Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{5n + 1}\)
4 Absolute Convergence · Level 2
Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n}{n^3 + 1}\)
5 Absolute Convergence · Level 2
Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin n}{2^n}\)
6 Absolute Convergence · Level 2
Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} \dfrac{n}{n^2 + 4}\)
7 Ratio Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n}{5^n} \)
8 Ratio Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{(-2)^n}{n^2} \)
9 Ratio Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} \dfrac{3^n}{2^n n^3} \)
10 Ratio Test · Level 3
\( \displaystyle\sum_{n=0}^{\infty} \dfrac{(-3)^n}{(2n + 1)!} \)
11 Ratio Test · Level 2
\( \displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k!} \)
12 Ratio Test · Level 2
\( \displaystyle\sum_{k=1}^{\infty} k e^{-k} \)
13 Ratio Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{10^n}{(n + 1) 4^{2n+1}} \)
14 Ratio Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{100^n} \)
15 Ratio Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n \pi^n}{(-3)^{n-1}} \)
16 Ratio Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n^{10}}{(-10)^{n+1}} \)
17 Ratio Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{\cos\left(n \dfrac{\pi}{3}\right)}{n!} \)
18 Ratio Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{n^n} \)
19 Ratio Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{n^{100} \cdot 100^n}{n!} \)
20 Ratio Test · Level 4
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{(2n)!}{(n!)^2} \)
21 Ratio Test · Level 4
\( 1 - \dfrac{2!}{1 \cdot 3} + \dfrac{3!}{1 \cdot 3 \cdot 5} - \dfrac{4!}{1 \cdot 3 \cdot 5 \cdot 7} + \cdots + (-1)^{n-1} \dfrac{n!}{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (2n - 1)} + \cdots \)
22 Ratio Test · Level 4
\( \dfrac{2}{3} + \dfrac{2 \cdot 5}{3 \cdot 5} + \dfrac{2 \cdot 5 \cdot 8}{3 \cdot 5 \cdot 7} + \dfrac{2 \cdot 5 \cdot 8 \cdot 11}{3 \cdot 5 \cdot 7 \cdot 9} + \cdots \)
23 Ratio Test · Level 4
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (2n)}{n!} \)
24 Ratio Test · Level 4
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n \dfrac{2^n n!}{5 \cdot 8 \cdot 11 \cdot \cdots \cdot (3n + 2)} \)
25 Root Test · Level 2
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{n^2 + 1}{2n^2 + 1}\right)^n\)
26 Root Test · Level 2
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-2)^n}{n^n}\)
27 Root Test · Level 3
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=2}^{\infty} \dfrac{(-1)^{n-1}}{(\ln n)^n}\)
28 Root Test · Level 3
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{-2n}{n + 1}\right)^{5n}\)
29 Root Test · Level 3
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \left(1 + \dfrac{1}{n}\right)^{n^2}\)
30 Root Test · Level 3
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=0}^{\infty} (\arctan n)^n\)
31 Absolute Convergence - Mixed · Level 3
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=2}^{\infty} \dfrac{(-1)^n}{\ln n}\)
32 Absolute Convergence - Mixed · Level 3
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{1 - n}{2 + 3n}\right)^n\)
33 Absolute Convergence - Mixed · Level 3
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-9)^n}{n \cdot 10^{n+1}}\)
34 Absolute Convergence - Mixed · Level 3
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n \cdot 5^{2n}}{10^{n+1}}\)
35 Absolute Convergence - Mixed · Level 4
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=2}^{\infty} \left(\dfrac{n}{\ln n}\right)^n\)
36 Absolute Convergence - Mixed · Level 3
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin\left(n \dfrac{\pi}{6}\right)}{1 + n \sqrt{n}}\)
37 Absolute Convergence - Mixed · Level 3
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n \arctan n}{n^2}\)
38 Absolute Convergence - Mixed · Level 3
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=2}^{\infty} \dfrac{(-1)^n}{n \ln n}\)
39 Ratio Test - Recursive · Level 3
The terms of a series are defined recursively by the equations \(a_1 = 2\), \(a_{n+1} = \dfrac{5n + 1}{4n + 3} a_n\) Determine whether \(\sum a_n\) converges or diverges.
40 Ratio Test - Recursive · Level 3
A series \(\sum a_n\) is defined by the equations \(a_1 = 1\), \(a_{n+1} = \dfrac{2 + \cos n}{\sqrt{n}} a_n\) Determine whether \(\sum a_n\) converges or diverges.
41 Absolute Convergence · Level 4
Let \({b_n}\) be a sequence of positive numbers that converges to \(\dfrac{1}{2}\). Determine whether the given series is absolutely convergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{b_n^n \cos n \pi}{n}\)
42 Absolute Convergence · Level 4
Let \({b_n}\) be a sequence of positive numbers that converges to \(\dfrac{1}{2}\). Determine whether the given series is absolutely convergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n n!}{n^n b_1 b_2 b_3 \cdots b_n}\)
43 Ratio Test - Concepts · Level 2
For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)?
(a) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^3}\)
(b) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{2^n}\)
(c) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-3)^{n-1}}{\sqrt{n}}\)
(d) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\sqrt{n}}{1 + n^2}\)

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44 Ratio Test · Level 4
For which positive integers \(k\) is the following series convergent? \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(n!)^2}{(k n)!}\)
45 Ratio Test - Proof · Level 4
(a) Show that \(\displaystyle\sum_{n=0}^{\infty} x^n / n!\) converges for all \(x\).
(b) Deduce that \(\operatorname*{lim}\limits_{n \rightarrow \infty} x^n / n! = 0\) for all \(x\).

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46 Remainder Estimates · Level 4
Let \(\sum a_n\) be a series with positive terms and let \(r_n = a_{n+1} / a_n\). Suppose that \(\operatorname*{lim}\limits_{n \rightarrow \infty} r_n = L < 1\), so \(\sum a_n\) converges by the Ratio Test. As usual, we let \(R_n\) be the remainder after \(n\) terms, that is, \(R_n = a_{n+1} + a_{n+2} + a_{n+3} + \cdots\)
(a) If \({r_n}\) is a decreasing sequence and \(r_{n+1} < 1\), show, by summing a geometric series, that \(R_n \leq \dfrac{a_{n+1}}{1 - r_{n+1}}\)
(b) If \({r_n}\) is an increasing sequence, show that \(R_n \leq \dfrac{a_{n+1}}{1 - L}\)

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47 Remainder Estimates · Level 3
(a) Find the partial sum \(s_5\) of the series \(\displaystyle\sum_{n=1}^{\infty} 1/(n 2^n)\). Use Exercise 46 to estimate the error in using \(s_5\) as an approximation to the sum of the series.
(b) Find a value of \(n\) so that \(s_n\) is within 0.00005 of the sum. Use this value of \(n\) to approximate the sum of the series.

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48 Remainder Estimates · Level 3
Use the sum of the first 10 terms to approximate the sum of the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{2^n}\) Use Exercise 46 to estimate the error.
49 Root Test - Proof · Level 5
Prove the Root Test. [Hint for part (i): Take any number \(r\) such that \(L < r < 1\) and use the fact that there is an integer \(N\) such that \(\sqrt[n]{|a_n|} < r\) whenever \(n \geq N\).]
50 Ratio Test - Applications · Level 4
Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula \(\dfrac{1}{\pi} = \dfrac{2 \sqrt{2}}{9801} \displaystyle\sum_{n=0}^{\infty} \dfrac{(4n)! (1103 + 26390n)}{(n!)^4 396^{4n}}\) William Gosper used this series in 1985 to compute the first 17 million digits of \(\pi\).
(a) Verify that the series is convergent.
(b) How many correct decimal places of \(\pi\) do you get if you use just the first term of the series? What if you use two terms?

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51 Absolute Convergence - Theory · Level 4
Given any series \(\sum a_n\), we define a series \(\sum a_n^+\) whose terms are all the positive terms of \(\sum a_n\) and a series \(\sum a_n^-\) whose terms are all the negative terms of \(\sum a_n\). To be specific, we let \(a_n^+ = \dfrac{a_n + |a_n|}{2}\), \(a_n^- = \dfrac{a_n - |a_n|}{2}\)
(a) If \(\sum a_n\) is absolutely convergent, show that both of the series \(\sum a_n^+\) and \(\sum a_n^-\) are convergent.
(b) If \(\sum a_n\) is conditionally convergent, show that both of the series \(\sum a_n^+\) and \(\sum a_n^-\) are divergent.

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52 Rearrangements - Proof · Level 5
Prove that if \(\sum a_n\) is a conditionally convergent series and \(r\) is any real number, then there is a rearrangement of \(\sum a_n\) whose sum is \(r\). [Hints: Use the notation of Exercise 51. Take just enough positive terms \(a_n^+\) so that their sum is greater than \(r\). Then add just enough negative terms \(a_n^-\) so that the cumulative sum is less than \(r\). Continue in this manner and use Theorem 11.2.6.]
53 Conditional Convergence · Level 5
Suppose the series \(\sum a_n\) is conditionally convergent.
(a) Prove that the series \(\sum n^2 a_n\) is divergent.
(b) Conditional convergence of \(\sum a_n\) is not enough to determine whether \(\sum n a_n\) is convergent. Show this by giving an example of a conditionally convergent series such that \(\sum n a_n\) converges and an example where \(\sum n a_n\) diverges.

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