**One another way that can evaluate **** by double integral is to write **** as follows**

**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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#
Month: October 2013

# The Basel Problem, Double Integral Method II

# The Basel Problem, Double Integral Method I

# A Beautiful Convergent Series

# Two Properties in the Elementary Set Theory

**One another way that can evaluate **** by double integral is to write **** as follows**

**This is true, since we have
**

**Now integrating by parts yields**

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

**Hence,**

**In 1644, the Italian mathematician Pietro Mengoli (1625-1686) posed the question: What’s the value of the sum**

**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**

**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**

**My first post is about two properties in the elementary set theory. You can see an amazing proof for them from me in the following.
**

**In proof I used the following property in logic**

**that is called “constructive dilemma”.**

**First Property:**

**Let** **, then I can write
**

**which implies that** **.**

**Second Property:**

**Let** **, then**

**which implies that** **.**