Answer
Converges to $0$
Work Step by Step
Consider $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{(\dfrac{10}{11})^n}{(\dfrac{9}{10})^n+(\dfrac{11}{12})^n}$
Since, $\lim\limits_{n \to \infty}x^n=0$
So, $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{(\dfrac{10}{11})^n}{(\dfrac{9}{10})^n+(\dfrac{11}{12})^n}= \lim\limits_{n \to \infty} \dfrac{(\dfrac{12}{11} \cdot \dfrac{10}{11})^n}{(\dfrac{12}{11} \cdot \dfrac{9}{10})^n+(\dfrac{12}{11} \cdot \dfrac{11}{12})^n}=\dfrac{0}{0+1}$
or, $=0$
Hence, $\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} is convergent and converges to $0$
