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Activity 8.2.4. The formulas we have derived for an infinite geometric series and its partial sum have assumed we begin indexing the sums at n = 0. If instead we have a sum that does not begin at n = 0, we can factor out common terms and use the established formulas. This process is illustrated in the examples in this activity. a. Consider the sum *********** 2 3 L® (²) * = m ( ² ) + ~ ( ² ) ² + (2) (²3) * + .... (2) (2) (2) k=1 Remove the common factor of (2) (3) from each term and hence find the sum of the series. b. Next let a and r be real numbers with -1 < r < 1. Consider the sum ∞ k=3 ark ar³ + ar¹ + ar5 +…... = Remove the common factor of ar³ from each term and find the sum of the series. k=n c. Finally, we consider the most general case. Let a and r be real numbers with −1 < r < 1, let n be a positive integer, and consider the sum ark = arn +arn+1 +arn+² +.... Remove the common factor of arn from each term to find the sum of the series.

Activity 8.2.4. The formulas we have derived for an infinite geometric series and its partial sum have assumed we begin indexing the sums at n = 0. If instead we have a sum that does not begin at n = 0, we can factor out common terms and use the established formulas. This process is illustrated in the examples in this activity. a. Consider the sum *********** 2 3 L® (²) * = m ( ² ) + ~ ( ² ) ² + (2) (²3) * + .... (2) (2) (2) k=1 Remove the common factor of (2) (3) from each term and hence find the sum of the series. b. Next let a and r be real numbers with -1 < r < 1. Consider the sum ∞ k=3 ark ar³ + ar¹ + ar5 +…... = Remove the common factor of ar³ from each term and find the sum of the series. k=n c. Finally, we consider the most general case. Let a and r be real numbers with −1 < r < 1, let n be a positive integer, and consider the sum ark = arn +arn+1 +arn+² +.... Remove the common factor of arn from each term to find the sum of the series.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Activity 8.2.4. The formulas we have derived for an infinite geometric series and its partial sum have assumed we begin indexing the sums at n = = 0. If instead we have a sum that does not begin at n = 0, we can factor out common terms and use the established formulas. This process is illustrated in the examples in this activity. a. Consider the sum ∞ k 2 3 Σ²) ( ³² ) * = (²) ( ² ) + (~² ( ² ) ² (2) ) + ( ( ² ) ² + (2) (2) 3 k=1 3 Remove the common factor of (2) (3) from each term and hence find the sum of the series. b. Next let a and r be real numbers with −1 < r < 1. Consider the sum ∞ Σark = ar³ + ar¹ + ar³ + k=3 Remove the common factor of ar³ from each term and find the sum of the series. c. Finally, we consider the most general case. Let a and r be real numbers with −1 < r < 1, let n be a positive integer, and consider the sum ∞ k=n ark = ar + ar ₂n+1 ₂n+2 + ar' + .. Remove the common factor of ar" from each term to find the sum of the series.
Transcribed Image Text:Activity 8.2.4. The formulas we have derived for an infinite geometric series and its partial sum have assumed we begin indexing the sums at n = = 0. If instead we have a sum that does not begin at n = 0, we can factor out common terms and use the established formulas. This process is illustrated in the examples in this activity. a. Consider the sum ∞ k 2 3 Σ²) ( ³² ) * = (²) ( ² ) + (~² ( ² ) ² (2) ) + ( ( ² ) ² + (2) (2) 3 k=1 3 Remove the common factor of (2) (3) from each term and hence find the sum of the series. b. Next let a and r be real numbers with −1 < r < 1. Consider the sum ∞ Σark = ar³ + ar¹ + ar³ + k=3 Remove the common factor of ar³ from each term and find the sum of the series. c. Finally, we consider the most general case. Let a and r be real numbers with −1 < r < 1, let n be a positive integer, and consider the sum ∞ k=n ark = ar + ar ₂n+1 ₂n+2 + ar' + .. Remove the common factor of ar" from each term to find the sum of the series.
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