It's neither obvious nor easy. It was an outstanding problem in the mid 1700's, and it was Euler that solved it but some superbly reckless arguments.

Euler's original paper:

http://www.17centurymaths.com/contents/euler/e041tr.pdf

The bottom of this page has a more modern treatment, although it takes a lot for granted:

http://www.math.wpi.edu/IQP/BVCalcHist/calc3.html

Another treatment of Euler's proof is here:

http://library.thinkquest.org/28049/Summing%20reciprocals.ht...