A coin is tossed until the first time a head turns up. If this occurs on the n th toss and n is odd you win 2n/n, but if n is even then you lose 2n/n. Then if your expected winnings exist they are given by the convergent series.

This is a series of variable harmonics that converges to log (2).

Step-by-step explanation:

Let's ignore for a moment that n is even or odd, and look at the expected value for any n. Let X be profit, which can be negative if we lose. The wait is given using (and assuming the coin is valid)

E[X]=∑n=1∞ ((-1) n+1*2∧n/n) ⋅1/2∧n=∑n=1∞ (-1) n+1*1/n.

This is a series of variable harmonics that converges to log (2). However, expectation exists only if it absolutely converges! Looking at

∑n=1∞∣ (-1) n+1*2∧n/n∣*1/2n=∑n=1∞1/n

we notice that a number of harmonics diverge, therefore, in fact, there is no expectation of X.