91Ó°ÊÓ

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{n}=n / 2^{n}$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{n}=n / e^{n}$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{k}=\frac{k^{e}}{e^{k}}$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{k}=\frac{\pi^{k}}{k^{\pi}}$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{n}=\left(\frac{1}{e}+\frac{1}{n}\right)^{n}$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{k}=\frac{1}{(1+\ln k)^{k}}$$

Use the root test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$a_{n}=\frac{(\ln (1+\ln n))^{n}}{(\ln n)^{n}}$$

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum_{k=1}^{\infty} a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$a_{k}=\frac{k !}{1 \cdot 3 \cdot 5 \cdots(2 k-1)}$$

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum_{k=1}^{\infty} a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$a_{k}=\frac{2 \cdot 4 \cdot 6 \cdots 2 k}{(2 k) !}$$

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum_{k=1}^{\infty} a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$a_{k}=\frac{1 \cdot 4 \cdot 7 \cdots(3 k-2)}{3^{k} k !}$$