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Infinite Geometric Series and Partial Sums - Graph and Analysis

Find and graph the partial sums of an infinite geometric series, and observe the pattern as n increases. Calculate sums of specific examples and determine if sums exist.

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Infinite Geometric Series and Partial Sums - Graph and Analysis

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  1. 1 1 1 Consider the infinite geometric series 1 + + + 16 8 4 2 1 + + . . . . Find and graph the partial sums Snfor n = 1, 2, 3, 4, and 5. Then describe what happens to Snas n increases. 32 S1 = = 0.5 + = 0.75 S2 = 1 1 2 4 0.88 1 S3 = + + 2 1 1 1 4 8 2 EXAMPLE 1 Find partial sums SOLUTION

  2. 1 1 1 1 1 1 1 + + 0.94 S4= + 4 2 2 4 8 8 16 1 1 + + S5 = + 0.97 + 16 32 EXAMPLE 1 Find partial sums From the graph, Snappears to approach 1as nincreases.

  3. a. . . . 5(0.8)i – 1 b. 1 – – + + 3 9 4 16 a1 a1 5 1 27 4 8 S = S = = = = = 25 1 – r 1 – r b. For this series, a1 = 1and r = – . ( ) 1 – 0.8 64 7 1 – i = 1 3 3 4 4 EXAMPLE 2 Find sums of infinite geometric series Find the sum of the infinite geometric series. SOLUTION a. For this series, a1 = 5 andr = 0.8.

  4. – 3 You know that a1 = 1 and a2 = – 3. So, r = – 3. 1 Because– 3 ≥ 1the sum does not exist. ANSWER The correct answer is D. EXAMPLE 3 Standardized Test Practice SOLUTION

  5. Consider the series + + + + + . . . . Find and graph the partial sumsSnforn = 1, 2, 3, 4 and 5. Then describe what happens to Snas nincreases. 4 2 25 5 8 16 125 625 32 3125 for Examples 1, 2 and 3 GUIDED PRACTICE Find the sums of the infinite geometric series.

  6. 2 3 2 5 for Examples 1, 2 and 3 GUIDED PRACTICE ANSWER S1 = S2 0.56 S3 0.62 S4 0.66 Snappearsto be approachingasnincreases.

  7. 8 8 3 3 3 . . . n – 1 n – 1 3. 2. 4. 3 + + 3 + + 16 4 64 n = 1 n = 1 – 1 5 2 4 no sum 2 ANSWER ANSWER 3 ANSWER 4 for Examples 1, 2 and 3 GUIDED PRACTICE Find the sum of the infinite geometric series, if it exists.

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