Convergence of $\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$

Question

I need help determining what following series converges to using the ratio test. $$\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$$ It's the end that really has me confused with what to do with the factorial.

Answer 1

Hint:

1) Write down the series for $e^3$. Terms are better than the summation notation. Note that this converges (Ratio Test).

2) Write down the series for $e^{-3}$.

3) Look.

The Ratio Test also settles the convergence of our series directly. Let $a_n=\dfrac{3^{2n}}{(2n)!}$. Then $\dfrac{a_{n+1}}{a_{n}}$ simplifies to $\dfrac{9}{(2n+1)(2n+2)}$.