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Chapter 7: Q. 84 (page 605)

Prove that if limk鈫�鈭�ak=L,then localid="1649337757642" limk鈫�鈭�ak+1=L

Short Answer

Expert verified

Hence proved thatlimk鈫�鈭�ak+1=L

Step by step solution

01

Step 1. Given information

The given sequencelimk鈫�鈭�ak=L

02

Step 2. The strategy to prove that limk→∞ak+1=L.

Use the defination of convergence of sequenceak

The sequenceak is convergent and convergers to L.

By the defination of convergence,for蔚>0 there is a positive integer N,such that

ak-L<蔚fork鈮�N

The result ak-L<蔚is true for all k鈮�N

Since the following inequality holds

k+1>k

Therefore

k+1>N(Becausek鈮�N)

Therefore,there is a positive integer Nsuch that

ak+1-L<蔚fork+1>N

Thus,limk鈫�鈭�ak+1=L

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Most popular questions from this chapter

Explain why a function a(x) has to be continuous in order for us to use the integral test to analyze a series 鈭�k=1鈭�akfor convergence.

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=0鈭�鈥�e鈭�k鈭�k=0鈭�鈥�e鈭�k

Determine whether the series 鈭�k=0鈭�-3k+14k-2converges 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.

鈭�k=1鈭�鈥�1+kk2

Given a series 鈭�k=1鈭�ak, in general the divergence test is inconclusive when ak鈫�0. For a geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.
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