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 £© The Sum of a Sequence
This section considers the problem of adding together the terms of a sequence. Of course,
this is a problem only if more than a finite number of terms of the sequence are nonzero.
In this case, we must decide what it means to add together an infinite number of nonzero
numbers. The first example shows how a relatively simple question may lead to such
infinite summations.
Example Suppose a game is played in which a fair coin is tossed until the first time
a head appears. What is the probability that a head appears for the first time on an
even-numbered toss? To solve this problem, we first need to determine the probability of
obtaining a head for the first time on any given even numbered toss, and then we need
to add all these probabilities together. Let Pn denote the probability that the first head
appears on the nth toss, n = 1, 2, 3, . . ... Then, since the coin is assumed to be fair,
1
P = .
1 2
Now in order to get a head for the first time on the second toss, we must toss a tail on the
first toss and then follow that with a head on the second toss. Since one-half of all first
tosses will be tails and then one-half of those tosses will be followed by a second toss of
heads, we should have
1 1 1
P2 = = .
2 2 4
Similarly, since one-fourth of all sequences of coin tosses will begin with two tails and then
half of these sequences will have a head for the third toss, we have
1 1 1
P3 = = .
4 2 8
Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3, . . .,
1
P = .
n 2n
Hence we have a sequence of probabilities {Pn} for n = 1, 2, 3, . . ., and, in order to find the
desired probability, we need to add up the even-numbered terms in this sequence. Namely,
the probability that a head appears for the first time on an even toss is given by
1 1 1
P + P + P + · · · = + + + · · ·