Stewart 9e Section 11.6: The Ratio and Root Tests

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Stewart 9e Section 11.6: The Ratio and Root Tests 0/55
1 Ratio Test - Conceptual · Level 1
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/Conditional 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/Conditional Convergence · Level 2
Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n}{5 n + 1}\)
4 Absolute/Conditional 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/Conditional 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/Conditional 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
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{5^n}\)
8 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-2)^n}{n^2}\)
9 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} \dfrac{3^n}{2^n n^3}\)
10 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=0}^{\infty} \dfrac{(-3)^n}{(2 n + 1)!}\)
11 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k!}\)
12 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{k=1}^{\infty} k e^{-k}\)
13 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{10^n}{(n+1) 4^{2 n + 1}}\)
14 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{100^n}\)
15 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n \pi^n}{(-3)^{n-1}}\)
16 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n^{10}}{(-10)^{n+1}}\)
17 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\cos\left(n \dfrac{\pi}{3}\right)}{n!}\)
18 Ratio Test · Level 3
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{n^n}\)
19 Ratio Test · Level 3
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n^{100} \cdot 100^n}{n!}\)
20 Ratio Test · Level 3
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(2 n)!}{(n!)^2}\)
21 Ratio Test · Level 3
Use the Ratio Test to determine whether the series is convergent or divergent. \(1 - \dfrac{2!}{1 \cdot 3} + \dfrac{3!}{1 \cdot 3 \cdot 5} - \dfrac{4!}{1 \cdot 3 \cdot 5 \cdot 7} + \ldots + (-1)^{n-1} \dfrac{n!}{1 \cdot 3 \cdot 5 \ldots (2 n - 1)} + \ldots\)
22 Ratio Test · Level 3
Use the Ratio Test to determine whether the series is convergent or divergent. \(\dfrac{2}{3} + \dfrac{2 \cdot 5}{3 \cdot 5} + \dfrac{2 \cdot 5 \cdot 8}{3 \cdot 5 \cdot 7} + \dfrac{2 \cdot 5 \cdot 8 \cdot 11}{3 \cdot 5 \cdot 7 \cdot 9} + \ldots\)
23 Ratio Test · Level 2
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{2 \cdot 4 \cdot 6 \ldots (2 n)}{n!}\)
24 Ratio Test · Level 3
Use the Ratio Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} (-1)^n \dfrac{2^n n!}{5 \cdot 8 \cdot 11 \ldots (3 n + 2)}\)
25 Root Test · Level 2
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{n^2 + 1}{2 n^2 + 1}\right)^n\)
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 2
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 2
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{-2 n}{n + 1}\right)^{5 n}\)
29 Root Test · Level 2
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 2
Use the Root Test to determine whether the series is convergent or divergent. \(\displaystyle\sum_{n=0}^{\infty} (\arctan n)^n\)
31 Absolute/Conditional Convergence · Level 2
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/Conditional Convergence · Level 2
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{1 - n}{2 + 3 n}\right)^n\)
33 Absolute/Conditional Convergence · Level 2
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/Conditional Convergence · Level 2
Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n \cdot 5^{2 n}}{10^{n+1}}\)
35 Absolute/Conditional Convergence · Level 2
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/Conditional Convergence · Level 2
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/Conditional Convergence · Level 2
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/Conditional Convergence · 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 Recursive Series · Level 3
The terms of a series are defined recursively by the equations \(a_1 = 2 \quad a_{n+1} = \dfrac{5 n + 1}{4 n + 3} a_n\) Determine whether \(\sum a_n\) converges or diverges.
40 Recursive Series · Level 3
A series \(\sum a_n\) is defined by the equations \(a_1 = 1 \quad a_{n+1} = \dfrac{2 + \cos n}{\sqrt{n}} a_n\) Determine whether \(\sum a_n\) converges or diverges.
41 Sequence-Dependent Series · Level 3
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 Sequence-Dependent Series · 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. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n n!}{n^n b_1 b_2 b_3 \ldots b_n}\)
43 Ratio Test - Inconclusive · Level 3
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) \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\sqrt{n}}{1 + n^2}\)

Enter your answer directly below each part above.

44 Ratio Test - Parameter · Level 3
For which positive integers \(k\) is the following series convergent? \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(n!)^2}{(k n)!}\)
45 Application of Ratio Test · Level 3
(a) Show that \(\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{n!}\) converges for all \(x\).
(b) Deduce that \(\operatorname*{lim}\limits_{n\rightarrow \infty} \dfrac{x^n}{n!} = 0\) for all \(x\).

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46 Error Estimate · 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. Let \(R_n\) be the remainder after \(n\) terms: \(R_n = a_{n+1} + a_{n+2} + a_{n+3} + \ldots\)
(a) If \(\{r_n\}\) is a decreasing sequence and \(r_{n+1} < 1\), show 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 Series Approximation · Level 3
(a) Find the partial sum \(s_5\) of the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n 2^n}\). 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 Series Approximation · 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 Proof - Root Test · Level 4
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 Ramanujan Series · Level 4
Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula \(\dfrac{1}{\pi} = \dfrac{2 \sqrt{2}}{9801} \displaystyle\sum_{n=0}^{\infty} \dfrac{(4 n)!(1103 + 26390 n)}{(n!)^4 396^{4 n}}\) 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 Positive and Negative Parts · 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 \(a_n^+ = \dfrac{a_n + |a_n|}{2} \quad a_n^- = \dfrac{a_n - |a_n|}{2}\) Notice that if \(a_n > 0\), then \(a_n^+ = a_n\) and \(a_n^- = 0\), whereas if \(a_n < 0\), then \(a_n^- = a_n\) and \(a_n^+ = 0\).
(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 Rearrangement Theorem · 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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54 Example - Ratio Test · Level 2
Test the convergence of the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n^n}{n!}\).
55 Example - Root Test · Level 2
Test the convergence of the series \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{2 n + 3}{3 n + 2}\right)^n\).

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