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

**I use the following amazing identity that you can see it’s proof in my previous posts
**

**By this identity, I can write**

**and this follows that
**

**Hence,**

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#
Series

# One Finite Sum

# Euler Sum

# Another Result By the Flajolet-Vardi Theorem

# A Beautiful Convergent Series II

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

**I use the following amazing identity that you can see it’s proof in my previous posts
**

**By this identity, I can write**

**and this follows that
**

**Hence,**

**Sums of the form** **where** **, sometimes are called Euler sums. There are several ways to evaluate these sums. A****about 240 years ago, Leonhard Euler(1707-1783) in 1775 proved that for ****,
**

**In the following you can see an elementary proof of this formula in the three steps:**

**Step 1:**

**Hence,**

**Step 2:**

**Now by recent formula that is also an alternate definition of the Harmonic numbers, we can write:**

**Step 3:**

**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** **,**

**In this post I want to generalize the given formula at my second post as follows**

**To prove this , I first prove following useful identity**

**PROOF: (Trigonometric Method) Note that
**

**Repeatedly applying** **, we arrive at the following formula:**

**but**

**and now note that **

**Because,**

**By using this recent identity we can write**