Chapter 4: Problem 1
Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{n}=1+(-1)^{n} \text { for } n \geq 1 $$
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Let \(a_{n}=2^{-\mid n / 2]}\) where \([x]\) is the greatest integer less than or equal to \(x\). Determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges and justify your answer.
The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.Sometimes the alternating series \(\sum_{n=1}^{\infty}(-1)^{n-1} b_{n}\) converges to a certain fraction of an absolutely convergent series \(\sum_{n=1}^{\infty} b_{n}\) a faster rate. Given that \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}\), find \(S=1-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\cdots .\) Which of the series \(6 \sum_{n=1}^{\infty} \frac{1}{n^{2}}\) and \(S \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}\) gives a better estimation of \(\pi^{2}\) using 1000 terms?
In the following exercises, use an appropriate test to determine whether the series converges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(n+1)}{n^{3}+3 n^{2}+3 n+1} $$
True or False? Justify your answer with a proof or a counterexample. $$ \text { If } \lim _{n \rightarrow \infty} a_{n} \neq 0 \text { , then } \sum_{n=1}^{\infty} a_{n} \text { diverges. } $$
In the following exercises, use an appropriate test to determine whether the series converges. $$ a_{k}=1 / 2^{\sin ^{2} k} $$
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