Answer
Converges
Work Step by Step
Let us consider $a_n=\dfrac{10^n}{( n+1)!}$ and $a_n$ refers to all positive values of $n$.
Here, $n \geq 10$ this implies that the sequence $u_n$ is not increasing.
Then, $\lim\limits_{n \to \infty} u_n=\lim\limits_{n \to \infty}\dfrac{10^n}{( n+1)!}=0$
Hence, the series converges by the Alternating Series Test.