properly speching, the expressions You have written are not 'simple series' butseries of functions. The have the general form...

(1)

Such series converges for in a given interval and and there are two types of convergence...

a) 'simple' convergence...

b)uniformconvergence...

The full explanation is a little complex and can't be done in short notes. I'm shure that You will be able to 'recovery' the time lost during your illness...

MHF is at Your disposition for that !...

Kind regards