**In this post I want to find the value of the sum
**

**Note that for** , **.**

**I use following formula that is called Gregory-Leibniz-Madhava’s series
**

**If we define** **function as follows:**

**then**

**Now I use this theorem**

**THEOREM (Flajolet–Vardi): If** **and** ** converges then,**

.

**PROOF:**

**Because** **,
**

**Hence, by Cauchy’s double series theorem, we can switch the order of summation:**

**This theorem implies that**

**and**