## Calculus (3rd Edition)

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