Question

S=1 - 1/4 + 1/6 - 1/9 + 1/11 - 1/14..........infinity. Find the sum of series.

Answer 1

`1+(1)/(6)+(1)/(11)+\cdots=\sum^(\infty)_(n=1) (1)/(5n-4) \\ \\ (1)/(4)+(1)/(9)+\cdots=\sum^(\infty)_(n=1) (1)/(5n-1) \\ \\ S=\sum^(\infty)_(n=1) \left((1)/(5n-4) -(1)/(5n-1) \right) \\ \\ =(\pi)/(5) \sqrt{(1)/(5) (5+2√(5))}`