I'm solving few math puzzles to train myself for a local math contest, and I'm stuck with this problem :

Compute the sum of :

$$\frac{1}{2*1} + \frac{1}{2*3} + \frac{1}{3*4} + ... + \frac{1}{2013*2014}$$

As said in the title, I know that all of those, can be written as : $\frac{1}{n(n+1)}$

But the $n(n+1)$ is not a geo seq nor an arithmetic one . So there are no theorems to compute their sum .

I'v also found a good number of similar puzzles so how can I solve this kind of problems ?


The $\frac{1}{n(n+1)} $ is called a telescopic sum, which telescopically can be expressed as : $\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$, which simplifies your initial expression. (This is the way that you express the series sum too).

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