<s>[INST] Finding the sum $\frac{1}{3\times5}+\frac{1}{5\times7}+\frac{1}{7\times9}+\cdots$ The sum of series is
$${1\over(3\times5)}+{1\over(5\times7)}+{1\over(7\times9)}+\cdots$$
I used to solve this problem as its $n$th term
$${1\over (2n+1)(2n+3)}$$
Now how can I proceed??
 [/INST] Hint : The $n$ th term of your series is $$ a_n= \dfrac{1}{2}\left ( \dfrac{1}{2n+1}- \dfrac{1}{2n+3}\right)$$
expanding your series you get $$ \dfrac{1}{2}\left(\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}...\right)$$
as you see the terms cancel out each other the only term left is $\dfrac{1}{3}$ with a factor of $\dfrac{1}{2}$ outside, giving your sum of the series as $\dfrac{1}{6}$
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