<s>[INST] Evaluating the Sum of $ \frac {1} {3}+\frac {1} {15}+\frac {1} {35}+\ldots +\frac {1} {4n^{2}-1} $ How would you evaluate the sum of this sequence ?
$$
\dfrac {1} {3}+\dfrac {1} {15}+\dfrac {1} {35}+\ldots +\dfrac {1} {4n^{2}-1}
$$
I realise the expression can be factorised but I can't really see what this can tell you.
 [/INST] $$\sum_{n=1}^{\infty }\frac{1}{4n^2-1}=\sum_{n=1}^{\infty }\frac{1}{2}\left[\frac{1}{2n-1}-\frac{1}{2n+1}\right]$$
$$=\frac{1}{2}\left[\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+....\right]=\frac{1}{2}$$
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