D' Alembert's Ratio test: convergence test

Question

Examine the convergence of the following two series

1. $$\sum_{n=1}^\infty \frac{n^4}{e^{n^2}}$$
2. $$\sum_{n=1}^\infty (-1)^{n+1}\frac{x^n}{1+x^n}$$

For the second, find the condition of convergence.

For the first, I want to use D' Alembert's Ratio test.

Let $u_n=\frac{n^4}{e^{n^2}}$ then $u_{n+1}=\frac{(n+1)^4}{e^{(n+1)^2}}$.

Now $$\frac{u_{n+1}}{u_n}=\frac{(n+1)^4\,e^{n^2}}{n^4\,e^{(n+1)^2}}=\frac{(1+1/n)^4}{e^{2n+1}}\to 0<1$$ as $n\to \infty$ So the 1st series converges. Correct?

Also help for the second

Answer 1

For the first your derivation is correct.

For the second note that

$$\left|\frac{x^{n+1}}{1+x^{n+1}}\frac{1+x^n}{x^n}\right|=\left|x\frac{1+x^n}{1+x^{n+1}}\right|\to L<1\iff |x|<1$$