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Chapter 7: Q. 21 (page 657)

Convergence or divergence of a series: For each of the series that follow, determine whether the series converges or diverges. Explain the criteria you are using and why your conclusion is valid.

∑k=1∞k3k

Short Answer

Expert verified

By ratio test, the sequence is convergent.

Step by step solution

01

Step 1. Given Information

The given sequence is∑k=1∞k3k.

02

Step 2. Apply the ratio test

  • The ratio test states that for the sequence ∑k=1∞ak, determine the value, p=limk→∞ak+1ak. If p<1, the series converges. If p>1, the series diverges. Otherwise, it is inconclusive.
  • Determine the value of p.

limk→∞ak+1ak=limk→∞k+13k+1k3k=limk→∞k+1k3k-k+1=limk→∞1+1k3-1=1+1∞3-1=119=19<1

  • Thus, by ratio test, the sequence is convergent.

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Most popular questions from this chapter

Determine whether the series ∑k=0∞4k+132kconverges or diverges. Give the sum of the convergent series.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

∑k=4∞ 2k−4k2−4k+3

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, R10.

(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 localid="1649224052075" Rn≤10-6.

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∑k=1∞ −12k , you can get the sum to be less than α. By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to α.)

Prove that if ∑k=1∞akconverges to L and ∑k=1∞bkconverges to M , then the series∑k=1∞ak+bk=L+M.
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