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# 5 #6 Show that it I an is divergent, then so is Shaw that it I lanti-and < & then convergent. must be the se Ž sequence an 1+ an hand

# 5 #6 Show that it I an is divergent, then so is Shaw that it I lanti-and < & then convergent. must be the se Ž sequence an 1+ an hand

Advanced Engineering Mathematics
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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# 5 # 6 7-1 Show that if I an must be convergent. is divergent, show that it I lanti-and < U = i then then so is the I sequence an hand
Transcribed Image Text:# 5 # 6 7-1 Show that if I an must be convergent. is divergent, show that it I lanti-and < U = i then then so is the I sequence an hand
Expert Solution
Step 1: Proof of the divergence

Given that ∑n=1∞an  is divergent 

 We know that

      an≤an+an2⇒an≤anan+1⇒anan+1≤an

Let bn=anan+1

We know  from comparison test that for ∑an and ∑bn  and ∑bn≤∑an hence, if ∑an converges, so does∑bn . But, if ∑andiverges, ∑bn may diverge or converge.

Consider an=1n the divergent series as we know ∑1n is divergent

Then

 bn=1n1+1n     =1n+1

∑1n+1 is  clearly divergent .

Therefore clearly ∑n=1∞anan+1 is clearly divergent for ∑n=1∞an  is divergent

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