So

Yes, that is a geometric sequence with "common ratio" which must be between -1 and 1 in order that it converge absolutely.I wasn't really sure how to approach it. I realized that it can become a convergent geometric series of |x+3| < 2, because that would make the absolute value of the r value = 1.

Okay, so that -2< x+ 3< 2, -5< x< -1.Taking this into account, I am proposing that -5< x < -1

I am afraid you have completely misunderstood this part of the question. It is asking for the sumI am pretty sure am I right up to this point after looking around online, but I am confused about the second part. I thought maybe you could find the infinite sum when x is -5 and when x is -1 and then subtract the first value from the second value. Is this the correct way to go about it? I got 8/21 as an answer.

Thanksas a function of x, not a difference between two values. You have correctly identified it as a geometric sequence, , with common ratio and first term a= 1. Now, recall that the sum of such a geometric series is .