Formula for the sum of fractions

Question

How to find the formula for the sum of fractions like this? $$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\ldots+\frac{1}{n\times (n+1)}=$$

This is a telescoping series. – Lucian Feb 15 '15 at 23:06 — Lucian, Feb 15 '15 at 23:06

score 2 · Answer 1 · answered Feb 15 '15 at 22:19

2

You can write $\dfrac 1{n(n+1)} = \dfrac 1n - \dfrac 1{n+1}$. You get a lot of cancellation after which the expression equals $1 - \dfrac{1}{n+1}$.

answered Feb 15 '15 at 22:19

Umberto P.

Did you seriously think that this question was never asked here before? – Feb 15 '15 at 22:21
2

To be honest I didn't give it a thought one way or another. Am I being naughty or something? – Umberto P. Feb 15 '15 at 22:22

score 1 · Answer 2 · answered Feb 15 '15 at 22:19

1

Since $$\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$$ for all $k$, you sum telescopes to $$1 - \frac{1}{n+1}.$$

answered Feb 15 '15 at 22:19

kobe

2 Answers2