121616949-math.247

121616949-math.247 - and more terms the sum gets closer and...

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10 Sequences and Series Consider the following sum: 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i + · · · 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
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Unformatted text preview: and more terms, the sum gets closer and closer to some ﬁxed value. That is, look at 1 2 = 1 2 3 4 = 1 2 + 1 4 7 8 = 1 2 + 1 4 + 1 8 15 16 = 1 2 + 1 4 + 1 8 + 1 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 2 + 1 4 + 1 8 + 1 16 + ··· + 1 2 i = 2 i-1 2 i = 1-1 2 i 233...
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