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 11−x under certain conditions on x, and differentiating the resulting inequality gives that ∑nxn−1 is convergent to the derivative of 11−x, 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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