**Let** **, then we have**

**and similarly** **.** **So we have**

**The sine function vanishes at integer multiples of** **, so it follows that** ** where** ** for all integers** **, that is, for** **for all** **. Thus,** **for**

**where we recall that** **. Since** **is strictly increasing on the interval** **, it** **follows that**

**moreover, since tangent is an odd function, we have** **for each** **. In particular we have found** **distinct roots of** **, so as a consequence of the fundamental theorem of algebra, we can write**

**So we have ****, and**

**I first prove that for any** **that is a power of** **and for any** **,**

**by using the following more familiar identity**

**we can easily arrive to the following identity that is valid for **** equal to any power of** **,**

**By considering what happens as** **in the recent formula, we can obtain that for** **a** **power of**

**and then I replace** **in** **and prove** **.**

**Now I choose** **and break up** **as follows**

**Since for** **,** **, we have
**

**and if I choose** **and** ** large enough such that** **, then**

**which implies that**

**Therefore from** **we have**

**Also we have**

**and once again** **by** **, we get**

**To summarize, I have shown that**

**Now taking** ** in** ** I arrive to**

**After rearrangement and then taking absolute values, we get**

**but since we have**

**I can write**

**Finally since** **,** **this inequality implies our goal, indeed**

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**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,**

**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:**

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**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** **,**

**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**

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**This is true, since we have
**

**Now integrating by parts yields**

**and by using the monotone convergence theorem, I can write
**

**Hence,**

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**First time, Leonhard Euler (1707-1783) in ****1735 proved that above series converges to ****. In this post you can see an easy proof by using double integral that published by Tom M. Apostol in 1983 in Mathematical Intelligencer.** **Apostol’s Proof:** **Note that**

**and by the monotone convergence theorem we get**

**We change variables in this by putting** **, so that** **. Hence**

**where** **is the square with vertices** **and** .

**Exploiting the symmetry of the square we get**

**Now** **, and if** ** then** **and** **. **

**It follows that** **and so** **. Hence**

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**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**

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