Find all such that the series

converges and compute the sum.

First, we apply the ratio test,

for all . Therefore, the series converges for all . Next, we compute the sum for ,

In the case then all of the terms except the term are 0, and that term is ; hence, the sum is .

How do we know that the first term is 1/2 (since the 0^0 part seems indeterminate)?

When you re-open the sum from n=2 back to n=0, iget that you are subtracting the terms corresponding no n=2 and n=1, in that case, shouldnt you have a (x^2)/2 term instead of a x term? Just wondering.

In the third line of the second equation block, right? I’m adding and subtracting the terms corresponding to and (adding the terms inside the sum, and the subtracting them outside the sum), not the term. That’s already in the sum. For the benefit of anyone looking at this later, I’ll expand on what happened in that line (it looks a bit terse in retrospect). The term is

and the term is

So then we have

Hopefully this makes sense.