Answer
converges
Work Step by Step
We are given the series$$\sum_{n=1}^{\infty} \frac{n^{10}+10^{n}}{n^{11}+11^{n}}$$
We use the limit comparison test with $\sum \left(\frac{10}{11}\right)^n $, which is a convergent geometric series with $|r| <1$
\begin{align*}
L&=\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&\\
&=\lim _{n \rightarrow \infty} \frac{n^{10}+10^{n}}{n^{11}+11^{n}} \left(\frac{11}{10}\right)^{n}\\
&=\lim _{n \rightarrow \infty} \frac{n^{10}+10^{n}}{10^{n}} \frac{11^{n}}{n^{11}+11^{n}}\\
&=\lim _{n \rightarrow \infty} \frac{\frac{n^{10}}{10^{n}}+1}{\frac{n^{11}}{11^{n}}+1}\\
&=1
\end{align*}
Thus, our starting series $\sum_{n=1}^{\infty} \frac{n^{10}+10^{n}}{n^{11}+11^{n}}$ also converges
