Here is my way of doing it.For now ignore the first term, that is for n=0. The second term is for n=1, etc

So you have for n=3 you have

So the general terms should be:

Oh, and of course there is the negative sign! Now if you plug in k = 0 in the above, you get 1 so we have taken care of the first term as well!

Now you need to test the series for convergence:

There are many ways you can test this for convergence and it should be pretty easy. I'm thinking comparison test with something with just a factorial on the bottom. Or you can compare successive terms.