Answer
converges to $1$.
Work Step by Step
As we know that a sequence converges when $\lim\limits_{n \to \infty}a_n$ exists.
Consider $a_n=1+(0.9)^n$
Apply limits to both sides.
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}[1+(0.9)^n]$
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}(1)+\lim\limits_{n \to \infty}(0.9)^n$
$\lim\limits_{n \to \infty}a_n=1+0$
$\lim\limits_{n \to \infty}a_n=1$
Therefore, the sequence converges to $1$.
