convergence divergence - Is the given series divergent /convergent?

Check the series whether it is convergent /divergent?

$$\sum\limits_{n=0}^{\infty}\frac{(3i)^{2n+1}}{(2n+1)!}$$

I was thinking about the Taylor series but could not get its,,,,how to expand

I think the series is divergent by D'Alembert ratio test.

Am I right? Can you verify it and tell the solution, please? I would be grateful.

Thanks in advance.

Answer

Not only does your series converge by the ratio test (see the previous two answers), it can be summed.

As
$$\sinh (z) = \sum_{n = 1}^\infty \frac{z^{2n + 1}}{(2n + 1)!}, \quad z \in \mathbb{C},$$
setting $z = 3i$ we have
$$\sum_{n = 0}^\infty \frac{(3i)^{2n + 1}}{(2n + 1)!} = \sinh (3i) = i \sin (3).$$