Solutions - 0200-01

(a)

$${\left|{\left(\frac{k}{3k+1}\right)}^k\right|}^{1/k}\gg \gg \frac{k}{3k+1}\stackrel{k\to \infty }{⟶}\frac{1}{3}=L<1$$

Therefore, by the root test the series converges absolutely.

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(b)

$$\begin{array}{c}|a_{n+1}\cdot a_n^{-1}|\gg \gg \frac{e^{n+1}}{(n+1)!}\cdot \frac{n!}{e^n}\\ \\ \gg \gg \frac{e}{n+1}\stackrel{n\to \infty }{⟶}0=L<1\end{array}$$

Therefore, by the ratio test the series converges absolutely.

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(c)

$$\begin{array}{c}|a_{n+1}\cdot a_n^{-1}|\gg \gg \frac{1}{(2(n+1))!}\cdot \frac{(2n)!}{1}\\ \\ \gg \gg \frac{1}{(2n+2)(2n+1)}\stackrel{n\to \infty }{⟶}0=L<1\end{array}$$

Therefore, by the ratio test the series converges absolutely.