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Magazine Problem 351
In the following exercises, use an appropriate test to determine whether the series converges. $$\sum_{n=1}^{\infty} \frac{(n-1)^{n}}{(n+1)^{n}}$$
Problem 352
In the following exercises, use an appropriate test to determine whether the series converges. $$a_{n}=\left(1+\frac{1}{n^{2}}\right)^{n}$$
Problem 353
In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=1 / 2^{\sin ^{2} k}$$
Problem 354
In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=2^{-\sin (1 / k)}$$
Problem 355
In the following exercises, use an appropriate test to determine whether the series converges. $$ a_{n}=1 /\left(\begin{array}{c} n+2 \\ n \end{array}\right) \text { where }\left(\begin{array}{l} n \\ k \end{array}\right)=\frac{n !}{k !(n-k) !} $$
Problem 356
In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=1 /\left(\begin{array}{l}2 k \\ k\end{array}\right)$$
Problem 357
In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=2^{k} /\left(\begin{array}{l}3 k \\ k\end{array}\right)$$
Problem 358
In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=\left(\frac{k}{k+\ln k}\right)^{k}$$
Problem 359
In the following exercises, use an appropriate test to determine whether the series converges. $$a_{k}=\left(\frac{k}{k+\ln k}\right)^{2 k}$$
Problem 362
The following series converge by the ratio test. Use summation \(\quad\) by \(\quad\) parts, $$ \sum_{k=1}^{n} a_{k}\left(b_{k+1}-b_{k}\right)=\left[a_{n+1} b_{n+1}-a_{1} b_{1}\right]-\sum_{k=1}^{n} b_{k+1}\left(a_{k+1}-a_{k}\right) $$ to find the sum of the given series. $$\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}}$$
