Stewart Section 11.5: Alternating Series

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Stewart Section 11.5: Alternating Series 0/34
1 Alternating Series - Concepts · Level 1
(a) What is an alternating series?
(b) Under what conditions does an alternating series converge?
(c) If these conditions are satisfied, what can you say about the remainder after \(n\) terms?

Enter your answer directly below each part above.

2 Alternating Series Test · Level 2
\( \dfrac{2}{3} - \dfrac{2}{5} + \dfrac{2}{7} - \dfrac{2}{9} + \dfrac{2}{11} - \cdots \)
3 Alternating Series Test · Level 3
\( -\dfrac{2}{5} + \dfrac{4}{6} - \dfrac{6}{7} + \dfrac{8}{8} - \dfrac{10}{9} + \cdots \)
4 Alternating Series Test · Level 2
\( \dfrac{1}{\ln 3} - \dfrac{1}{\ln 4} + \dfrac{1}{\ln 5} - \dfrac{1}{\ln 6} + \dfrac{1}{\ln 7} - \cdots \)
5 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{3 + 5n} \)
6 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^{n+1}}{\sqrt{n + 1}} \)
7 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n \dfrac{3n - 1}{2n + 1} \)
8 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n \dfrac{n^2}{n^2 + n + 1} \)
9 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n e^{-n} \)
10 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n \dfrac{\sqrt{n}}{2n + 3} \)
11 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} (-1)^{n+1} \dfrac{n^2}{n^3 + 4} \)
12 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} (-1)^{n+1} n e^{-n} \)
13 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} e^{\dfrac{2}{n}} \)
14 Alternating Series Test · Level 2
\( \displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} \arctan n \)
15 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=0}^{\infty} \dfrac{\sin\left(n + \dfrac{1}{2}\right) \pi}{1 + \sqrt{n}} \)
16 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=0}^{\infty} \dfrac{n \cos n \pi}{2^n} \)
17 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n \sin\left(\dfrac{\pi}{n}\right) \)
18 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n \cos\left(\dfrac{\pi}{n}\right) \)
19 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n \dfrac{n^n}{n!} \)
20 Alternating Series Test · Level 3
\( \displaystyle\sum_{n=1}^{\infty} (-1)^n (\sqrt{n + 1} - \sqrt{n}) \)
21 Alternating Series - Estimation · Level 3
Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-0.8)^n}{n!}\)
22 Alternating Series - Estimation · Level 3
Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. \(\displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} \dfrac{n}{8^n}\)
23 Alternating Series - Estimation · Level 3
Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n^6}\) \((|\text{error}| < 0.00005)\)
24 Alternating Series - Estimation · Level 3
Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\displaystyle\sum_{n=1}^{\infty} \dfrac{\left(-\dfrac{1}{3}\right)^n}{n}\) \((|\text{error}| < 0.0005)\)
25 Alternating Series - Estimation · Level 3
Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^2 2^n}\) \((|\text{error}| < 0.0005)\)
26 Alternating Series - Estimation · Level 3
Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? \(\displaystyle\sum_{n=1}^{\infty} \left(-\dfrac{1}{n}\right)^n\) \((|\text{error}| < 0.00005)\)
27 Alternating Series - Approximation · Level 3
Approximate the sum of the series correct to four decimal places. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n}{(2n)!}\)
28 Alternating Series - Approximation · Level 3
Approximate the sum of the series correct to four decimal places. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n^6}\)
29 Alternating Series - Approximation · Level 3
Approximate the sum of the series correct to four decimal places. \(\displaystyle\sum_{n=0}^{\infty} (-1)^n n e^{-2n}\)
30 Alternating Series - Approximation · Level 3
Approximate the sum of the series correct to four decimal places. \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n 4^n}\)
31 Alternating Series - Concepts · Level 2
Is the 50th partial sum \(s_{50}\) of the alternating series \(\displaystyle\sum_{n=1}^{\infty} (-1)^{n-1} / n\) an overestimate or an underestimate of the total sum? Explain.
32 Alternating Series - p-values · Level 3
For what values of \(p\) is the series convergent? \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^p}\)
33 Alternating Series - p-values · Level 3
For what values of \(p\) is the series convergent? \(\displaystyle\sum_{n=1}^{\infty} \dfrac{(-1)^n}{n + p}\)
34 Alternating Series - p-values · Level 4
For what values of \(p\) is the series convergent? \(\displaystyle\sum_{n=2}^{\infty} (-1)^{n-1} \dfrac{(\ln n)^p}{n}\)

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