How to estimate summations

i.e.

Is there a way you can know this solution is 2 without having to perform all of the calculations I did to find which number the sums are approaching? And is there a general method for questions like these to find the solution without having to perform a lot of calculations?

Re: How to estimate summations

Quote:

Originally Posted by

**daigo** i.e.

Is there a way you can know this solution is 2 without having to perform all of the calculations I did to find which number the sums are approaching? And is there a general method for questions like these to find the solution without having to perform a lot of calculations?

Yes, this is an infinite geometric series with , so the sum is .

Re: How to estimate summations

The sum of a geometric series is given by

, given . In this case, .