Chapter 7: Q. 56 (page 626)

Use the divergence test to prove that a p-series $\sum_{k = 1}^{\infty} \frac{1}{k^{p}}$ diverges when p < 0.

Short Answer

Expert verified

$The expression \frac{1}{k^{p}} is zero for values of p which are equal to or greater than 1 . To make the expression \frac{1}{k^{p}} non - zero, the value of p must be less than 0 . Thus, by divergence test, the series \sum_{k = 1}^{\infty} \frac{1}{k^{p}} is divergent for p < 0 .$

Step by step solution

Step 1. Given information is:

$\sum_{k = 1}^{\infty} \frac{1}{k^{p}}$

Step 2. Finding the limit:

$The value of the sequence {a_{k}} = \{\frac{1}{k^{p}}\} is : \lim_{k \to \infty} a_{k} = \lim_{k \to \infty} \frac{1}{k^{p}} The expression \frac{1}{k^{p}} is zero for values of p which are equal to or greater than 1 . The series \sum_{k = 1}^{\infty} \frac{1}{k^{p}} is divergent only when \lim_{k \to \infty} a_{k} does not converge to zero .$

Step 3. Result

$To make the expression \frac{1}{k^{p}} non - zero, the value of p must be less than 0 . Thus, by divergence test, the series \sum_{k = 1}^{\infty} \frac{1}{k^{p}} is divergent for p < 0 .$

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Use the divergence test to prove that a p-series $\sum_{k = 1}^{\infty} \frac{1}{k^{p}}$ diverges when p < 0.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Finding the limit:

Step 3. Result

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Use the divergence test to prove that a p-series∑k=1∞1kp diverges when p < 0.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Finding the limit:

Step 3. Result

One App. One Place for Learning.

Most popular questions from this chapter

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Use the divergence test to prove that a p-series $\sum_{k = 1}^{\infty} \frac{1}{k^{p}}$ diverges when p < 0.