Solutions - 0200-02

(a)

$$\begin{array}{c}|a_{n+1}\cdot a_n^{-1}|\gg \gg \frac{2^{n+1}}{{(n+1)}^{100}}\cdot \frac{n^{100}}{2^n}\\ \\ \gg \gg 2{\left(\frac{n}{n+1}\right)}^{100}\stackrel{n\to \infty }{⟶}2=L>1\end{array}$$

Since $L>1$, the ratio test says that the series diverges.

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(b)

$$\sqrt[n]{|a_n|}=\frac{5n}{10n+4}\stackrel{n\to \infty }{⟶}\frac{5}{10}=L<1$$

Since $L<1$, the root test says that the series converges.

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(c)

$$\begin{array}{c}|a_{n+1}\cdot a_n^{-1}|=\frac{\sqrt{n+1}}{3^{n+1}}\cdot \frac{3^n}{\sqrt{n}}\\ \\ \gg \gg \frac{1}{3}\sqrt{\frac{n+1}{n}}\stackrel{n\to \infty }{⟶}\frac{1}{3}=L<1\end{array}$$

Since $L<1$, the ratio test says that the series converges.