Find Limit of 1/2 Series: 1/(n^2 + n)

[moo5003]

Homework Statement

What is the limit of 1/2 Series from (n=1 to n=Infinity) of 1/(n^2 + n).

The Attempt at a Solution

This is a simplification of finding the integral of an oscillating function from 0 to 1 that makes triangles of height one between 1/n and 1/(n+1)

Thus the area of each triangle is 1/2 * (1/n - 1/(n+1)) = 1/(2n^2 + 2n)

The integral should therefore be equal to the above given limit. From intuition I believe the answer should be 1/2, though I would appreciate any help in determining how this limit is found. Seems like it should be relativley easy to find.

 

[Avodyne]

Well, it's easier in the original form. You want to sum 1/n - 1/(n+1) from n=1 to infinity. Let's write out the first few terms:
$$\left({1\over 1}-{1\over 2}\right)<br /> +\left({1\over 2}-{1\over 3}\right)<br /> +\left({1\over 3}-{1\over 4}\right)+\ldots$$
Notice any way to simplify this?

 

[Feldoh]

As Avodyne said, it's a telescoping series