Example 4.15. Consider the series $\sum_{n=1}^{\infty}\left(\frac{7+(-1)^{n}}{2}\right)^{-n}$ and $\sum_{n=1}^{\infty}\left(\frac{7+(-1)^{n}}{2}\right)^{n}$
Solution. Theorem 4.11 is not applicable to both these series since the limit $\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}$ does not exist. However, applying Theorem 4.13 yields certain results:

$$
\varlimsup_{n \rightarrow \infty} \sqrt[n]{\left(\frac{7+(-1)^{n}}{2}\right)^{-n}}=\frac{1}{3}1
$$

therefore the series $\sum_{n=1}^{\infty}\left(\frac{7+(-1)^{n}}{2}\right)^{n}$ diverges.

For the considered series, the D'Alembert's ratio test cannot be applied even in its strengthened version (Theorem $4.10)$, since

$$
\begin{aligned}
& \varlimsup_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\infty \\
& \underline{l i m}_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=0
\end{aligned}
$$

where $a_{n}=\left(\frac{7+(-1)^{n}}{2}\right)^{n}$.