a. Explanation Given: The geometric series f ( r ) = ∑ k = 0 ∞ r k , where | r | < 1 . Result used: The geometric series ∑ n = 1 ∞ a r n − 1 = a + a r + a r 2 + ⋯ is convergent if | r | < 1 and its sum is a 1 − r , where a ≠ 0 is the first term and r is the common ratio of the series. Calculation: Obtain the geometric series f ( r ) = ∑ k = 0 ∞ r k at r = − 0.9 . Here, the geometric series ∑ k = 0 ∞ ( − 0.9 ) k with first term of the series is a = 1 and common ratio is | r | = 0.9 < 1 . Compute the sum of the geometric series by the Result stated above as follows. ∑ k = 0 ∞ ( − 0.9 ) k = 1 1 + 0.9 = 1 1.9 ≈ 0.526 That is, ∑ k = 0 ∞ ( − 0.9 ) k ≈ 0.526 . Obtain the geometric series f ( r ) = ∑ k = 0 ∞ r k at r = − 0.7 . Here, the geometric series ∑ k = 0 ∞ ( − 0.7 ) k with first term of the series is a = 1 and common ratio is | r | = 0.7 < 1 . Compute the sum of the geometric series by the Result stated above as follows, ∑ k = 0 ∞ ( − 0.7 ) k = 1 1 + 0.7 = 1 1.7 ≈ 0.588 That is, ∑ k = 0 ∞ ( − 0.9 ) k ≈ 0.526 . Obtain the geometric series f ( r ) = ∑ k = 0 ∞ r k at r = − 0.5 . Here, the geometric series ∑ k = 0 ∞ ( − 0.5 ) k with first term of the series is a = 1 and common ratio is | r | = 0.5 < 1 . Compute the sum of the geometric series by the Result stated above as follows, ∑ k = 0 ∞ ( − 0.5 ) k = 1 1 + 0.5 = 1 1.5 ≈ 0.667 That is, ∑ k = 0 ∞ ( − 0.5 ) k ≈ 0.667 . Obtain the geometric series f ( r ) = ∑ k = 0 ∞ r k at r = − 0.2 . Here, the geometric series ∑ k = 0 ∞ ( − 0.2 ) k with first term of the series is a = 1 and common ratio is | r | = 0.2 < 1 . Compute the sum of the geometric series by the Result stated above as follows, ∑ k = 0 ∞ ( − 0.2 ) k = 1 1 + 0.2 = 1 1.2 ≈ 0.833 That is, ∑ k = 0 ∞ ( − 0 b. To determine To sketch: The graph of f ( r ) = ∑ k = 0 ∞ r k with | r | < 1 . c. To determine To evaluate: The value of lim r → 1 − f ( r ) and lim r → 1 + f ( r ) .

2nd Edition

ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 8.3, Problem 99E

To determine

To find: The value of the series f(r) for various values of r.

To determine

To sketch: The graph of f(r)=∑k=0∞rk with |r|<1.

To determine

To evaluate: The value of limr→1−f(r) and limr→1+f(r).

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Chapter 8.3, Problem 98E

Chapter 8.3, Problem 100E

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The value of the series f ( r ) for various values of r .

Solution Summary: The author explains the value of the geometric series f(r) for various values of r.

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To find: The value of the series f(r) for various values of r.

To sketch: The graph of f(r)=∑k=0∞rk with |r|<1.

To evaluate: The value of limr→1−f(r) and limr→1+f(r).

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