They are using the power series expansion of :
The given sum is .
We can combine the " " and " " terms to get
However, the answer to your question depends upon what A is. The problem you gives says "Using your result from part (a)" but you don't tell us what "part (a)" is or what your result was. Without that information, it is impossible to go further.
In your second question, you are given . Integrating by parts, letting and , we have and so that the integral becomes
The "suitable rigor" (did they really mispell "suitable" in your book?) referred to means take the limits to be "0" and "A" and then taking the limit as A goes to infinity: for all n> 0. That makes the integral .
(In what you have they have used " " instead of my "A".)