Hi, I could use some help on part (b) of the following question from my Complex Variables, book. I didn't know if this belonged in the Calculus forum or not.
(a) Use the identity
(b) Use the above identity to also establish
Check the answer by differentiating the series from (a)
Part (a) was easy:
The check in part (b) is easy, too. But I can't do part (b) using the identity given. I tried squaring the identity, but this is as far as I've gotten (essentially, I just eliminated the negative sign):
I've also got this (but I don't know if it's worth anything, or if it is even correct):
I have answered part (a) and the check in part (b) of the following question. But I cannot figure out part (b)
Expand the left summation (but not the right one):
Distribute the right summation:
Expand each summation on a row (lining them up nice would make the following step easier to see, too):
Now, draw a line underneath and add like terms vertically (you know, like you did when you first learned addition back in the first grade) and you get:
So the coefficient of each term is one more than the exponent, giving the summation on the right. Which is what you said.