Consider the following sum:
1 1 1 1 1 + + + + + + 2 4 8 16 ··· 2i ···
The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite sum? While at first it may seem difficult or impossible, we have certainly done something similar when we talked about one quantity getting “closer and closer” to a fixed quantity. Here we could ask whether, as we add more and more terms, the sum gets closer and closer to some fixed value. That is, look at
1 1 = 2 2 3 1 1 = + 4 2 4 7 1 1 1 = + + 8 2 4 8 15 1 1 1 1 = + + + 16 2 4 8 16
and so on, and ask whether these values have a limit. It seems pretty clear that they do, namely 1. In fact, as we will see, it’s not hard to show that
1 1 1 1 1 2i 1 1 + + + + + = − =1 2 4 8 16 ··· 2i 2i − 2i
253 254 Chapter 11 Sequences and Series
and then 1 lim 1 =1 0=1. i→∞ − 2i − There is one place that you have long accepted this notion of infinite sum without really thinking of it as a sum: 3 3 3 3 1 0.33333¯ = + + + + = , 10 100 1000 10000 ··· 3 for example, or 1 4 1 5 9 3.14159 ... = 3 + + + + + + = π. 10 100 1000 10000 100000 ··· Our first task, then, to investigate infinite sums, called series, is to investigate limits of sequences of numbers. That is, we officially call ∞ 1 1 1 1 1 1 = + + + + + + 2i 2 4 8 16 ··· 2i ··· Xi=1 a series, while 1 3 7 15 2i 1 , , , ,..., − ,... 2 4 8 16 2i is a sequence, and ∞ 1 2i 1 = lim − , 2i i→∞ 2i Xi=1
that is, the value of a series is the limit of a particular sequence.