**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:**

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