180 Days of Ideas for Discussion in Math Class.
(as of 9July2014, we're in overtime!)
Rewrite 1/1 as -1/2 + 3/2. The sequence 3/2 - 5/6 + … + 19/90 is(-1)^(n+1)(2n + 1)/(n(n+1)), n = 1,2,…,9. Take differences pairwise, then add the results pairwise, then do it again to find the eight terms sum to 8/9 = 80/90. Doing the algebra and finding the final answer is left as an exercise for the mathematically literate.