Let {an}n21 be a sequence of positive numbers such that E an diverges. For n 2 1, let s,= a1+...+ an.(1) Prove that the seriesanEnzlant1diverges.(2) Prove that for all N 21 and all n > 1,nSNAN+k> 1SN+kSN+nk=1anDeduce that the series diverges.Sn(3) Prove that for all n 2 2,An11Sn-1SnDeduce that the series z converges.VI

Answered: Let {an}n21 be a sequence of positive…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let {an}n21 be a sequence of positive numbers such that E an diverges. For n 2 1, let s,= a1
+...+ an.
(1) Prove that the series
an
Enzlant1
diverges.
(2) Prove that for all N 21 and all n > 1,
n
SN
AN+k
> 1
SN+k
SN+n
k=1
an
Deduce that the series diverges.
Sn
(3) Prove that for all n 2 2,
An
1
1
Sn-1
Sn
Deduce that the series z converges.
VI

Expert Solution

Step 1

Note: According to Bartleby guidelines; for more than one question asked, only the first one is to be answered.

Given that $\sum_{n \geq 1} a_{n}$ where $a_{n}$ is the sequence of positive numbers.

Then since $\{a_{n}\}$ is a sequence of positive numbers and since $\sum_{n \geq 1} a_{n}$ is divergent it follows that $\{a_{n}\}$ is an increasing sequence. Hence it follows that $\lim_{n \to \infty} \frac{1}{a_{n}} = 0$

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