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Infinite Geometric Series. Section 9.2. If you start adding the terms of the arithmetic sequence 1, 2, 3, . . . , you get larger and larger values. Even if the terms of an arithmetic sequence are small, as in 0.001, 0.002, 0.003, . . . , the partial sums eventually get large.
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Infinite Geometric Series Section 9.2
If you start adding the terms of the arithmetic sequence 1, 2, 3, . . . , you get larger and larger values. Even if the terms of an arithmetic sequence are small, as in 0.001, 0.002, 0.003, . . . , the partial sums eventually get large. • As the number of terms increases, the magnitude of the partial sum increases. But consider the geometric sequence 0.4, 0.04, 0.004, 0.0004, . . . . • It has common ratio 1/10 , so the terms get smaller. The partial sums seem to follow a pattern.
If you sum infinitely many terms of this sequence, would the result be infinitely large? • It appears that the partial sums will not get infinitely large; they are all less than 0.5. The indicated sum of a geometric sequence is a geometric series. • An infinite geometric series is a geometric series with infinitely many terms. In this lesson you will specifically look at convergent series, for which the sequence of partial sums approaches a long-run value as the number of terms increases.
Example A • Jack baked a pie and promptly ate one-half of it. Determined to make the pie last, he then decided to eat only one-half of the pie that remained each day. • Record the amount of pie eaten each day for the first seven days. • For each of the seven days, record the total amount of pie eaten since it was baked. • If Jack lives forever, then how much of this pie will he eat?
The amount of pie eaten each day is a geometric sequence with first term 1/2 and common ratio 1/2. • The first seven terms of this sequence are • Find the partial sums, S1 through S7, of the terms in part a.
It may seem that eating pie “forever” would result in eating a lot of pie. However, if you look at the pattern of the partial sums, it seems as though for any finite number of days Jack’s total is slightly less than 1. This leads to the conclusion that Jack would eat exactly one pie in the long run. This is a convergent infinite geometric series with long-run value 1.
Recall that a geometric sequence can be represented with an explicit formula in the form un = u1 ● r n-1 or un =u0 ●r n, where r represents the common ratio between the terms. The investigation will help you create an explicit formula for the sum of a convergent infinite geometric series.
Infinite Geometric Series Formula • In algebra you may have learned a method for writing a repeating decimal as a fraction. This method can also help you find the value of an infinite geometric series. • For example, consider the repeating decimal 0.44444. . . . It can be thought of as the sum of the infinite geometric series 0.4+0.04+ 0.004 + . . . . • If S =0.44444. . . , then 10S = 4.44444. . . . Subtract 10S -S to eliminate the decimal portion:
Step 1: Consider the sequence 0.4, 0.04, 0.004, . . . underlying the series S. Identify the first term, u1, and the common ratio, r, of the sequence. How is the multiplier 10 (in the expression 10S) derived from r? What other multipliers could be used to eliminate the repeating decimal portion? • Step 2: Use the common ratio, r, as the multiplier instead of 10 and solve for S again.Is your answer equivalent to 4/9 ?
Step 3: Consider the sequence 0.9, 0.09, 0.009, . . . . Identify the first term, u1, and the common ratio, r. Now use the method from Step 2 to find the sum S of the series 0.9 0.09 0.009 . . . . • Step 4: Repeat Step 3 for the sequence 0.27, 0.0027, 0.000027, . . . . Remember to use r as the multiplier. • Step 5: Repeat Step 3 for the series u1 + r •u1 + r2 •u1 + r3 ·u1 + . . . , assuming that it has sum S. Create a new series with sum r S. Then subtract to find a formula for S based on u1 and r. • Use a variety of r-values, including both positive and negative numbers, to create several geometric sequences. Look at the partial sums of each sequence as n gets very large. Use your formula from Step 5 to help you describe when the partial sums of a geometric sequence will converge to a unique number S. Use your examples to justify your answer.
Example B • Consider an ideal (frictionless) ball bouncing after it is dropped. The distances in inches that the ball falls on each bounce are represented by 200, 200(0.8), 200(0.8)2, 200(0.8)3, and so on. Summing these distances creates a series. Find the total distance the ball falls during infinitely many bounces.
Example C • Consider the infinite series • Express this sum of infinitely many terms as a decimal. • Identify the first term, u1, and the common ratio, r. • Express the sum as a ratio of integers.
