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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.

鈭�k=1鈭�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鈭�k=1鈭�k3k.

02

Step 2. Apply the ratio test

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

limk鈫�鈭�ak+1ak=limk鈫�鈭�k+13k+1k3k=limk鈫�鈭�k+1k3k-k+1=limk鈫�鈭�1+1k3-1=1+1鈭�3-1=119=19<1

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

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