The expressions n and n+1 are both real, so you can bring them outside of the absolute value signs.
As for finding the sum, if you differentiate the geometric series term-by-term, you get . I think that will help.
- Hollywood
I've attached a picture on the problem and my progress so far.
What confuses me is the i in the denominator. I'm guessing the limit is just 1 / (1 - i). If so, |z|(1 - i) > 1 gives me the radius of convergence. This does not look like a geometric series, so I'm not sure how to find the sum...
The expressions n and n+1 are both real, so you can bring them outside of the absolute value signs.
As for finding the sum, if you differentiate the geometric series term-by-term, you get . I think that will help.
- Hollywood
Thank you! So, taking the derivative of this series won't get me anywhere, so I'm guessing I have to take the anti-derivative, find the sum of that, and finally take the derivative of the result.
However, when taking the antiderivative, I just got stuck again. Maybe I did the antiderivative wrong, or there's something more that could be done to make the expression into a geometric series. I've attached a picture of the antiderivative I got.
Nothing at all wrong with the words - nothing wrong with "integrate" or "integral" either. I was just thinking that this is not the type of problem where we typically use those words.
Upon further reflection, however, perhaps "antiderivative" is actually the perfect word for this situation....
- Hollywood