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I'm trying to simplify the following:

$\sum_{i = 1}^{n} \frac{1}{(2i-1)(2i+1)}$

into

$\frac{n}{2n+1}$

How should I proceed?

  • 3
    Hint: Telescopic sums. $$\frac{1}{(2i-1)(2i+1)} = \frac{1}{2}\left(\frac{1}{2i-1} - \frac{1}{2i+1}\right)$$ – 2017-01-16

1 Answers 1

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Notice that

$$\frac1{(2i-1)(2i+1)}=\frac12\left(\frac1{2i-1}-\frac1{2i+1}\right)$$

This gives us the following telescoping series:

$$\sum_{i=1}^n\frac1{(2i-1)(2i+1)}=\sum_{i=1}^n\frac12\left(\frac1{2i-1}-\frac1{2i+1}\right)=\frac12\left(1-\frac1{2n+1}\right)=\frac n{2n+1}$$