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Chapter 7: Q. 93 (page 593)

Prove that ifakk=1鈭� is a sequence of positive real numbers, then the sequence Snn=1鈭�, where the sequence Sn = a1 + a2 +路路路+an, is an increasing sequence.

Short Answer

Expert verified

Proved.

Step by step solution

01

Step 1. Given

Consider the sequencesakk=1鈭�.

02

Step 2. Proof

ThesequenceSn=a1+a2+a3+...+an......(1)Changenton+1inequation(1)Sn+1=a1+a2+a3+...+an+an+1.....(2)Subtract(1)from(2)Sn+1-Sn=a1+a2+a3+...+an+an+1-(a1+a2+a3+...+an)=a.HenceSn+1-Sn=an+1SinceSn+1鈮�SnHencethesequenceisincreasingsequence.

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Most popular questions from this chapter

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�k1+kk2

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

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�1k2

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.

If limk鈫�鈭�akbkWhereLis a positive finite number, what may we conclude about the two series?
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