I'm trying to determine the convergence of this

Jayleen Aguirre

Jayleen Aguirre

Answered question

2022-04-08

I'm trying to determine the convergence of this series:
n=1(12·34·56·...2n-32n-2·2n-12n)a

Answer & Explanation

stecchiniror7

stecchiniror7

Beginner2022-04-09Added 14 answers

Let
an=1234562n32n2
Note that an+1=an2n+12(n+1)
We will show that for all n:
14nan12n+1
Having established this, it will follow by comparison with p-series that n=1ana
converges if and only if a>2
We establish the inequalities by induction.
Towards showing that an12n+1 first note that a1121+1
Now suppose an12n+1. Then
an+12[2n+12(n+1)]212n+1=2n+14(n+1)2=2n+14n2+8n+4
2n+14n2+8n+1=12n+1
So, an12n+1 for all n
Towards showing that an14n. Then an+12[2n+12(n+1)]2·14n. Now
[2n+12(n+1)]214n14(n+1)=116n(n+1)2>0
thus
an+1214(n+1)
So, an14n for all n.

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