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Geometric Series (page 4 of 5) Sections: Terminology and notation, Basic examples, Arithmetic and geometric sequences, Arithmetic series, Finite and infinite geometric series You can take the sum of a finite number of terms of a geometric sequence. And, for reasons you'll study in calculus, you can take the sum of an infinite geometric sequence, but only in the special circumstance that the common ratio r is between –1 and 1; that is, you have to have | r | < 1. For a geometric sequence with first term a1 = a and common ratio r, the sum of the first n terms is given by: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
In the special case that | r | < 1, the infinite sum exists and has the following value:
Note: Your book may have a slightly different form of the partial-sum formula above. For instance, the "a" may be multiplied through the numerator, the factors in the fraction might be reversed, or the summation may start at i = 0 and have a power of n + 1 on the numerator. All of these forms are equivalent, and the formulation above may be derived from polynomial long division.
The first few terms are –6, 12, –24, so this is a geometric series with common ratio r = –2. The first term is a = –6. Plugging into the summation formula, I get:
So the value of the summation is 2 097 150 I need to find the sum of the first ten terms. The first term is a = 250. Dividing pairs of terms, I get 100 ÷ 250 = 2/5, 40 ÷ 100 = 2/5, etc, so r = 2/5. Then:
They've given me S4 and r. Since there is a common ratio, I know this must be a geometric series, so, plugging into the series-sum formula, I get:
Multiply on both sides by 27/40 to solve for the first term a = a1, I get:
Then:
There's a trick to this. I first have to break the repeating decimal into separate terms: 0.333... = 0.3 + 0.03 + 0.003 + 0.0003 + ... This shows the repeating pattern explicitly: For each term, I have a decimal point, followed by a steadily-increasing number of zeroes, and then a "3". This can be written in fractional form, and then converted into geometric-series form:
Then 0.333... is an infinite geometric series with a = 3/10 and r = 1/10. Since | r | < 1, I can use the formula for summing infinite geometric series:
Using the summation formula to show that the geometric series "expansion" of 0.333... has a value of one-third is the "showing" that the exercise asked for. You can use this method to convert any repeating decimal to its fractional form:
First I'll break this into its constituent parts, so I can find the pattern: 1.363636.. = 1 + 0.36 + 0.0036 + 0.000036 + ... There are two digits that repeat, so the fractions are a little bit different. But this is still geometric:
Then this is "1" plus a geometric series having a = 9/25 and r = 1/100. Then the sum is:
Note: This technique can also be used to convert any repeating decimal into fractional form, and also can be used to prove that 0.999... = 1. << Previous Top | 1 | 2 | 3 | 4 | 5 | Return to Index
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Copyright © 2006-2008 Elizabeth Stapel | About | Terms of Use |
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