Find the variance in the problem of Exercise 27 for independent trials.

Exercise 27

We run a sequence of trials for which the probabilities of success are different. Namely, for the first trial, it is 1/2, for the second, it is 1/4, and so on: for the kth trial, it is 1/2^k. Find the expected number of successes in n trials. Consider the case n = ∞.