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Chapter 7: Q. 14 (page 656)

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

鈭�k=0鈭�3-25k

Short Answer

Expert verified

The sum of the series is157.

Step by step solution

01

Step 1. Given Information

The given series is鈭�k=0鈭�3-25k.

02

Step 2. Determine if a geometric series 

  • Find the ratio of the consecutive terms.

3-25k+13-25k=-25k+1-k=-25

  • Since the ratio of the consecutive terms is same, the given series is a geometric series with a common ratio r=-25.
03

Step 3. Determine the convergence 

  • Determine the first term and the common ratio.

c=3-250=3r=-25

  • If r<1, the series converges to c1-r.

c1-r=31--25=31+25=157

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Most popular questions from this chapter

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1,鈭�)such that role="math" localid="1649081384626" limx鈫�鈭�f(x)=伪>0. What can the divergence test tell us about the series 鈭�k=1鈭�f(k)?

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.

鈭�k=1鈭�鈥�1+kk2

Find an example of a continuous function f :[1,鈭�)鈫�Rsuch that 鈭�1鈭�fxdxdiverges and localid="1649077247585" 鈭�k=1鈭�fkconverges.

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

鈭�k=1鈭�鈥�tan鈭�1鈦�k1+k2

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 localid="1649224052075" Rn鈮�10-6.

鈭�k=0鈭�e-k
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