Introduction We have seen series that are finite, meaning they have a limited number of terms, but what happens to a series that has infinite terms? A geometric series that has infinite terms (that is, where n = ∞), is called an infinite geometric series. In an infinite geometric series, summing all the terms in the series is not an easy task, because the series starts with n = 1 and goes on until infinity. Do such sums exist? As you will see in this lesson, for infinite geometric series with certain values of the common ratio r, a sum does exist. 1 2.5.3: Sum of an Infinite Geometric Series
Transcript
IntroductionWe have seen series that are finite, meaning they have a limited number of terms, but what happens to a series that has infinite terms? A geometric series that has infinite terms (that is, where n = ∞), is called an infinite geometric series. In an infinite geometric series, summing all the terms in the series is not an easy task, because the series starts with n = 1 and goes on until infinity. Do such sums exist? As you will see in this lesson, for infinite geometric series with certain values of the common ratio r, a sum does exist.
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2.5.3: Sum of an Infinite Geometric Series
Key Concepts• Whether an infinite geometric series has a sum
depends on whether the absolute value of its common ratio, r, is greater or less than 1, and on the behavior of the partial sums of the series.
• Partial sums are the sums of part of a series. They are the result of adding pairs of terms. Partial sums are used to determine if a series converges or diverges.
• In a series, to converge means to approach a finite limit. If the sequence of the partial sums of a series approaches the value of a given number (the limit), then the entire series converges to that limit. In other words, the series has a sum.
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2.5.3: Sum of an Infinite Geometric Series
Key Concepts, continued• An infinite series converges when the absolute value
of the common ratio r is less than 1 • If the series does not have a sum—that is, the
sequence of its partial sums does not approach a finite limit—then the series is said to diverge.
• In a series that diverges, the absolute value of the common ratio, r, is greater than 1
• Recall that a series cannot have an r value equal to 1 (r ≠ 1), because this would result in a 0 in the denominator of the sum formula. Fractions with a denominator of 0 are undefined.
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2.5.3: Sum of an Infinite Geometric Series
Key Concepts, continued
• Consider the sequence . The terms and
the partial sums of the series from n = 1 to n = 8 are
calculated in the tables that follow.
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2.5.3: Sum of an Infinite Geometric Series
Key Concepts, continued
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2.5.3: Sum of an Infinite Geometric Series
Terms and partial sums for
(continued)
Key Concepts, continued
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2.5.3: Sum of an Infinite Geometric Series
Key Concepts, continued• Each term in the sequence is evaluated, then added
to the value of the previous term. Notice that the partial sum of each pair of terms grows closer to 0.5 with each additional term.
• However, even as the partial sums approach 0.5, the values of the specific terms being added are growing smaller: a8 is less than a7, which is less than a6, and so on. Since the values being added keep shrinking, it is impossible to reach the exact value 0.5.
• However, if it were possible to sum this series to infinity, presumably the series would sum to 0.5.
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2.5.3: Sum of an Infinite Geometric Series
Key Concepts, continued
• The sum formula for an infinite geometric series, if
the sum exists, is given by , where Sn is the
sum, a1 is the first term, and r is the common ratio.
This formula is derived from the sum formula for a
finite geometric series.
• If the series converges—that is, if —then the sum exists and this formula can be used to find it.
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2.5.3: Sum of an Infinite Geometric Series
Key Concepts, continued
• Recall that the summation notation for a finite
geometric series is , where n is the number of
terms in the sequence. • An infinite series in summation notation has an infinity
symbol ( ) above the sigma instead of n to indicate an unlimited number of terms:
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2.5.3: Sum of an Infinite Geometric Series
Common Errors/Misconceptions• mistaking a series that diverges for a series that
converges • forgetting to multiply by the reciprocal when dividing
fractions
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice
Example 2Determine the sum, if a sum exists, of the following geometric series.
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 2, continued
1. Determine if the series has a sum.
Examine the value of r. If , then the series has a
sum. In this series, . We can see that the
absolute value of r is less than 1: . Therefore,
the series has a sum.
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 2, continued
2. Determine what variables found in the sum formula for an infinite geometric series are known and unknown. The sum of an infinite geometric series is given by
Known variable:
Unknown variable: a1
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 2, continued
3. Determine the value of a1 in the series.
Given series
Write out the first term and
substitute 1 for k.
Simplify.
The first term of the series is 4.
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 2, continuedThe geometric series is given by a • r n – 1. The first term has an n value of 1, which makes the power 0. Anything raised to a power of 0 is 1. This means that you will be multiplying the value of a by 1, which just results in a.
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 2, continued
4. Substitute the variables into the sum formula for an infinite geometric series.
Example 3Determine the sum, if a sum exists, of the following geometric series. Justify the result.
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 3, continued
1. Determine if the series has a sum.
Examine the value of r. If , then the series has a
sum. In this series, , which simplifies to . We
can see that the absolute value is greater than 1;
therefore, this series does not have a sum.
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 3, continued
2. Justify the result.
Since the value of n in the series goes to
infinity, r n – 1 grows without bound. To see this, make a
table of values, as shown on the following slides.
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 3, continued
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2.5.3: Sum of an Infinite Geometric Series
(continued)
Terms and partial sums for
Guided Practice: Example 3, continued
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2.5.3: Sum of an Infinite Geometric Series
Guided Practice: Example 3, continuedWe can see that the partial sums do not converge to a limit; they increase in value infinitely. Since the partial sums do not converge, this series is diverging and has no sum. The result found in step 1 is justified.
