# What is the formula for $1/(1\cdot 2)+1/(2\cdot 3)+1/(3\cdot 4)+\ldots +1/(n(n+1))$

How can I find the formula for the following equation?

$$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\ldots +\frac{1}{n(n+1)}$$

More importantly, how would you approach finding the formula? I have found that every time, the denominator number seems to go up by n+2, but that's about as far as I have been able to get:

$1/2 + 1/6 + 1/12 + 1/20 + 1/30...$ the denominator increases by $4,6,8,10,12...$ etc.

So how should I approach finding the formula? Thanks!

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If you simplify your partial sums, you get $\frac12,\frac23,\frac34,\frac45,....$ Does this give you any ideas?

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Thanks! This was very helpful :) Now I just gotta prove it by induction! – Charles Jan 24 '13 at 18:31

Hint: Use the fact that $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ and find $S_n=\sum_1^n\left(\frac{1}{k}-\frac{1}{k+1}\right)$.

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Simple and nice (+1) – user 1618033 Jan 24 '13 at 18:59
@Chris'ssister: Thanks a lot for your consideration. – Babak S. Jan 24 '13 at 19:03
Nicely said, Babak(+1)! – amWhy Jan 24 '13 at 19:21

While exploitation of the resulting telescoping series after partial fraction expansion is a very simple way forward, I thought it might be instructive to present another way forward. Here, we write

\begin{align} \sum_{k=1}^N \frac{1}{k(k+1)}&=\sum_{k=1}^\infty \int_0^1y^{k-1}\,dy \int_0^1 x^k\,dx\\\\ &=\int_0^1\int_0^1x\sum_{k=1}^N (xy)^{k-1}\,dx\\\\ &=\int_0^1\int_0^1 x\frac{1-(xy)^N}{1-xy}\,dx\,dy\\\\ &=\int_0^1\int_0^x \frac{1-y^N}{1-y}\,dy\,dx\\\\ &=\int_0^1\int_y^1\frac{1-y^N}{1-y}\,dx\,dy\\\\ &=\int_0^1(1-y^N)\,dy\\\\ &=1-\frac1{N+1} \end{align}

as expected!

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