**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**

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