Let $t\in \mathbb{R}^{*}, t\neq 1$ and $S_k=t^k+\frac{1}{t^k}$, for every $k\in \mathbb{N}^{*}$.
If $x_n=\frac{1}{S_1}+\frac{1}{S_2}+...+\frac{1}{S_n}$ then prove that $(x_n)_{n\geq 1}$ is convergent.
If $t>0$ then $(x_n)_n$ is increasing and $S_k>\frac{1}{2^{k-1}}(t+\frac{1}{t})^k$, for every $k\geq 1$, by Jensen's inequality.
Then $\frac{1}{2}\cdot\frac{1}{S_k}<[\frac{1}{2}(t+\frac{1}{t})]^k$. But $\frac{1}{2} (t+\frac{1}{t})>1$. So $(x_n)_{n\geq 1}$ is bounded, therefore is convergent.
I dont't know to solve the case when $t<0$.
Note that $$\frac{1}{|S_k|}=\frac{1}{|t|^k+1/|t|^k}\leq \frac{1}{\max\{|t|^k,1/|t|^k\}}=\frac{1}{(\max\{|t|,1/|t|\})^k}=q^k$$ where $q:=\min\{|t|,1/|t|\}<1$. So the series $$\sum_{k=1}^{\infty}1/S_k=\lim_{n\to\infty}x_n$$ is absolutely convergent and therefore convergent.