**By using the Flajolet-Vardi theorem we can find the value of the another amazing convergent series. Indeed, following series **

**I first recall the Flajolet-Vardi theorem that you can find it’s proof in my second post:**

**Flajolet-Vardi Theorem:**

**If** **and** **converges then,**

**This theorem shows that** **, because if we let** **, then** **and by this theorem **

**Therefore now we must find the value of ****. we use the Taylor expansion of** **and the fact that the sum** **is** **if** **divides** **and** **otherwise.**

**There are some** **‘s of complex numbers. Those numbers have always non-negative real part, for the** **Argument** **we take the angle between** **and** **, so that it fits with the power series for** **. **

**but also**

**We thus have**

**We have to take the limit** **. The** **term disappears, so we get**

**Examples:**

**For** **we have**

**and also for** **,**