Euler numbers generating sequence radius of convergence

I'm not sure which section a post on finding the Euler numbers using series belongs, so I put it here.

Find (a) the radius of convergence, and (b) the first six Euler numbers from:

I haven't any idea how to start the part (a).

I can't put the finishing touches on part (b):

So,

However, I can't quite get this to a format where I can get the Euler numbers.

I know I can't flip the bottom fraction since it's an infinite sum. But I only need the first six, yet I still can't do it. The first one I can do with the first term of the series (right?):

and (so the back of my book says).

The next term, and the one after that, is:

Now what? How do I find the coefficient's of z, z^2, etc. from this?

I'm still working on part (a), so I don't have a specific question yet.

Thanks.

P.S. How do you make a new line in this forum? I tried \\, \\*, and \newline with no success and had to make separate 'math' tags for each line.