Find closed formula for $a_{n+1}=(n+1)a_{n}+n!$ $a_{n+1}=(n+1)a_{n}+n!$ where a0=0 and n>=0.
To get the closed form, I'm trying to find an exponential generating function for the above recurrence, but it doesn't seem to be very nice. Am I going about this the wrong way? If so, are there any helpful tricks I can use to solve this recurrence for a(n) explicitly? Thanks.

Dividing by $(n+1)!$ gives us that $$\frac{a_{n+1}}{(n+1)!}=\frac{a_{n}}{n!}+\frac{1}{n+1}$$
Now substituting $\frac{a_{n}}{n!}=b_{n}$ $$b_{n+1}=b_{n}+\frac{1}{n+1}$$
Thus $b_{n}$ is is the harmonic series. 
