Q) find the sum of the series -




solution : the first thing to do is to find the general term of this series.
it is easy to see that the denominators are all perfect squares.
and the numerators are consecutive odd numbers starting from .
the numerator is given by by the simple formula for arithmetic progression.
for the denominators, we have to consider one more thing. all of them are the product of two consecutive perfect squares.

so, they are given by .

hence

so, the sum is nothing but as tends to infinity.

which is .