Wolfram Alpha has made even simpler, and (more correct proof)
http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-....
http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-....
http://www.wolframalpha.com/input/?i=1+-+2+%2B+3+-+4+%2B+5+-....
"http://www.wolframalpha.com/input/pdfGet.jsp?id=MSP55619821e...