Proving Convergence of Infinite Series with Changing Signs

[ait.abd]

Homework Statement

Show the convergence of the series
$$\sum_{n=1}^{inf}(\frac{1}{n}-\frac{1}{n+x})$$
of real-valued functions on $$R - \{-1, -2, -3, ...\}$$.

Homework Equations

The Attempt at a Solution

I first thought of solving this using telescoping series concept but it didn't work out. Also, I tried to prove it by saying that since denominator goes $$n^2$$ it should converge but what I doubt is that the elements of the series change sign and the modulus of its terms will not be lower than $$1/n^2$$.

 

[micromass]

Well, you could try to turn this in a telescoping series.

Pick m a natural number such that x<m. Then

$$\frac{1}{n}-\frac{1}{n+x}\leq \frac{1}{n}-\frac{1}{n+m}$$...

 

[ait.abd]

Well, you could try to turn this in a telescoping series.

Pick m a natural number such that x<m. Then

$$\frac{1}{n}-\frac{1}{n+x}\leq \frac{1}{n}-\frac{1}{n+m}$$...

Thanks micromass!

 

[ait.abd]

I have a slight problem with the solution micromass hope you can clarify. I was looking at the statement of comparison tests that states that $$a_n, b_n > 0$$ for comparison test to be valid whereas in the solution above we can have negative individual terms as well? Is it so or I am looking at it wrongly?
e.g.
$$n=1, x=-0.1 \implies 1-10/9 < 0$$

 

[samii]

Given the series:

1/1 + 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4) + ...

How would u find the sum of the series?
Any help pleaseee!