1 Dec 2008

Geometric Series and their Sums in Edwards & Penney

presentation of Edwards & Penney's work, by by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

The series

is said to be a geometric series if each term after the first is a fixed multiple of the term immediately before it. That is, there is a number r, called the ratio of the series, such that

Each successive term would be an instance of the ratio multiplied by itself, so every geometric series takes the form:

for the first n + 1 terms as the nth partial sum of the series.

But the summation begins at n = 0 (instead of n = 1). So it is more convenient to render the sum (which the text names Eq 5):

For example, the infinite series:

This is a geometric series whose first term is a = 2 and whose ratio is r = 1/3.

The Sum of a Geometric Series:

For example:

Below we see a graphic representation of the partial sums of this series approaching its sum of 3/5, alternating above it and below it at decreasing differences:

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p694a-695.

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