Chapter 7: Q. 57 (page 626)

Use the divergence test to prove that a geometric series $\sum_{k = 1}^{\infty} c r^{k}$ diverges when $|r| \geq 1$ and $c \neq 0 .$

Short Answer

Expert verified

$The series does not converge to 0 . Therefore, the series is divergent by divergence test . Thus, by divergence test, the geometric series \sum_{k = 1}^{\infty} {cr}^{k} diverges when |r| \geq 1 and c \neq 0 .$

Step by step solution

Step 1. Given information is:

$\sum_{k = 1}^{\infty} c r^{k} a n d |r| \geq 1, c \neq 0$

Step 2. Examining different cases for c>0

$First, assume c > 0 Case 1 : when the ratio r \leq - 1 The ratio of geometric series \sum_{k = 1}^{\infty} {cr}^{k} is r \leq - 1, then the value of \lim_{k \to \infty} {cr}^{k} does not exist . Case 2 : When the ratio r = 1 The ratio of geometric series \sum_{k = 1}^{\infty} {cr}^{k} is r = 1, then the series become \sum_{k = 1}^{\infty} {cr}^{k} = \sum_{k = 1}^{\infty} c Therefore, \lim_{k \to \infty} {cr}^{k} = c The sequence does not converge to 0 . Therefore, for r = 1, the series is divergent by divergence test . Case 3 : When the ratio r > 1 The value of \lim_{k \to \infty} {cr}^{k} is : \lim_{k \to \infty} {cr}^{k} = \infty Therefore, the sequence does not converge to 0 .$

Step 3. Result

$Therefore, the series is divergent by divergence test . In the similar fashion, result is proved for c < 0 . Thus, by divergence test, the geometric series \sum_{k = 1}^{\infty} {cr}^{k} diverges when |r| \geq 1 and c \neq 0 .$

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Use the divergence test to prove that a geometric series $\sum_{k = 1}^{\infty} c r^{k}$ diverges when $|r| \geq 1$ and $c \neq 0 .$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Examining different cases for c>0

Step 3. Result

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Use the divergence test to prove that a geometric series ∑k=1∞crkdiverges whenr≥1 and c≠0.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Examining different cases for c>0

Step 3. Result

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Use the divergence test to prove that a geometric series $\sum_{k = 1}^{\infty} c r^{k}$ diverges when $|r| \geq 1$ and $c \neq 0 .$