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Chapter 7: Q. 57 (page 626)

Use the divergence test to prove that a geometric series 鈭k=1鈭crkdiverges whenr鈮1 and c鈮0.

Short Answer

Expert verified

Theseriesdoesnotconvergeto0.Therefore,theseriesisdivergentbydivergencetest.Thus,bydivergencetest,thegeometricseries鈭k=1鈭crkdivergeswhenr鈮1andc鈮0.

Step by step solution

01

Step 1. Given information is:

鈭k=1鈭crkandr鈮1,c鈮0

02

Step 2. Examining different cases for c>0

First,assumec>0Case1:whentheratior鈮-1Theratioofgeometricseries鈭k=1鈭crkisr鈮-1,thenthevalueoflimk鈫鈭crkdoesnotexist.Case2:Whentheratior=1Theratioofgeometricseries鈭k=1鈭crkisr=1,thentheseriesbecome鈭k=1鈭crk=鈭k=1鈭cTherefore,limk鈫鈭crk=cThesequencedoesnotconvergeto0.Therefore,forr=1,theseriesisdivergentbydivergencetest.Case3:Whentheratior>1Thevalueoflimk鈫鈭crkis:limk鈫鈭crk=鈭Therefore,thesequencedoesnotconvergeto0.

03

Step 3. Result

Therefore,theseriesisdivergentbydivergencetest.Inthesimilarfashion,resultisprovedforc<0.Thus,bydivergencetest,thegeometricseries鈭k=1鈭crkdivergeswhenr鈮1andc鈮0.

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Most popular questions from this chapter

Let a:[1,鈭)鈫鈩be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral 鈭1鈭a(x)dxconverges, then the series localid="1649180069308" 鈭k=1鈭a(k)does too.

Let 鈭k=0鈭crkand鈭k=0鈭bvkbe two convergent geometric series. Prove that 鈭k=0鈭crk.bvk converges. If neither c nor b is 0, could the series be 鈭k=0鈭crk鈭k=0鈭bvk?

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak鈫0, then 鈭k=1鈭akconverges.

(b) True or False: If 鈭k=1鈭akconverges, then ak鈫0.

(c) True or False: The improper integral 鈭1鈭f(x)dxconverges if and only if the series 鈭k=1鈭f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series 鈭k=1鈭k-pconverges.

(f) True or False: If f(x)鈫0as x鈫鈭, then 鈭k=1鈭f(k) converges.

(g) True or False: If 鈭k=1鈭f(k)converges, then f(x)鈫0as x鈫鈭.

(h) True or False: If 鈭k=1鈭ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

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1kk

Given thata0=-3,a1=5,a2=-4,a3=2and鈭akk=2鈭=7, find the value of鈭akk=4鈭.
See all solutions

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