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Chapter 7: Q. 15 (page 656)

Geometric series: For each of the series that follow, find the sum or explain why the series diverges.

鈭�k=0鈭�1k

Short Answer

Expert verified

The given series diverges.

Step by step solution

01

Step 1. Given Information

The given series is鈭�k=0鈭�1k.

02

Step 2. Determine if a geometric series  

  • Find the ratio of the consecutive terms.

1k+11k=1k+1-k=1

  • Since the ratio of the consecutive terms is same, the given series is a geometric series with a common ratio r=1.
03

Step 3. Determine the convergence  

  • Determine the first term and the common ratio.

c=10=1r=1

  • If r<1, the series converges.
  • So, the given series diverges.

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Most popular questions from this chapter

Determine whether the series 鈭�k=0鈭�e蟺3kconverges or diverges. Give the sum of the convergent series.

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.

鈭�k=1鈭�鈥�k!kk

For each series in Exercises 44鈥�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.

鈭�k=2鈭�1k(lnk)2

Determine whether the series 鈭�n=4鈭�411nconverges or diverges. Give the sum of the convergent series.

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

鈭�k=2鈭�1klnkp
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