Convergence question from Stromberg Classical Analysis I have been trying to find whether the following series converge. If so, to what limit?

Define $x_0=0$ and $x_1=1$, and
   $$  
x_{n+1~}=\frac{1}{n+1}x_{n-1}+\frac{n}{n+1}x_n \qquad n\geq 1
 $$

There are other ways to show convergence, but I showed that $x_{2n-2}\leq x_{2n}\leq x_{2n+1}\leq x_{2n-1}$ for all $n$ and $\lim_{n\to\infty}(x_{2n-1}-x_{2n})=0$. Thus, $\lim_{n\to\infty} x_n=x$.
To find the actual value of $x_n$ I have tried many things but could not find the value. I think the most promising manipulation I got so far is the following
$$
\frac{1}{2}+\frac{1}{3!}-\frac{1}{4!}+\frac{1}{5!}-\frac{1}{6!}+\frac{1}{7!}-\frac{1}{8!}+\dots
$$
Thanks for any help!

It seems to me that your guessed limit is
$$
1 + \sinh(1) - \cosh(1).
$$
Just compare the expansions
$$
\cosh(x)=\sum_{n=1}^\infty \frac{x^{2n}}{(2n)!}, \quad \sinh(x) = \sum_{n=1}^\infty \frac{x^{2n+1}}{(2n+1)!}.
$$
You should try to prove it, maybe by induction.
