Thursday, June 30, 2016

sequences and series - How to compute this infinite sum?





I'm trying to compute the infinite sum



n=1n(12)n



which I believe should represent the expected amount of coin flips needed to get a head. Can someone remind me how to do this?


Answer




The key is that the infinite sum xn converges to 11x under certain conditions on x, and differentiating the resulting inequality gives that nxn1 is convergent to the derivative of 11x, under the same conditions. Multiplying this by x gives the sum nxn, which is the result you are looking for with x=12, which does fall under the set for which the first equality holds.


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