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Chapter 7: Q. 31 (page 657)

Check the convergence $\sum_{k = 1}^{\infty} {(- 1)}^{k} \ln k$

Short Answer

Expert verified

The series diverges.

Step by step solution

01

Step 1. Given

The given series $\sum_{k = 1}^{\infty} {(- 1)}^{k} \ln k$

02

Step 2. Checking of convergence

The general term of the series $\sum_{k = 1}^{\infty} {(- 1)}^{k} \ln k$ is $a_{k} = \ln k$

localid="1649739242985" $The value of \lim_{k \to \infty} a_{k} i s : \lim_{k \to \infty} a_{k} = \lim_{k \to \infty} \ln k = \infty Since \lim_{k \to \infty} a_{k} \emptyset 0 Hence the series \lim_{k \to \infty} \ln k diverges .$

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Most popular questions from this chapter

Provide a more general statement of the integral test in which the function f is continuous and eventually positive, and decreasing. Explain why your statement is valid.

In Exercises 48–51 find all values of p so that the series converges.

$\sum_{k = 1}^{\infty} \frac{k}{{(1 + k^{2})}^{p}}$

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

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

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ converges.

For each series in Exercises 44–47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder $R_{10}$ .

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that.

$\sum_{k = 2}^{\infty} \frac{1}{k {(\ln k)}^{2}}$

See all solutions

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