Closed Formula for Geometric Series and Sums

I just have two questions regarding closed form formulas for geometric series and summing infinite series.

For the closed form formula, I have an expression for the term:

But I am not sure how to convert this to closed form since I am not really sure what closed form looks like. Is it just to convert it into a finite sum equation like this:

For the infinite series, I have the formula:

Where

So the very first term would be:

And this would go in the numerator, but I am not sure what would go in the denominator. I really appreciate everyone's help.