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

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I&#039;m trying to determine the convergence of this series:∑n=1∞(12·34·56·...2n-32n-2·2n-12n)a

stecchiniror7 · Accepted Answer

Letan=12⋅34⋅56⋅…⋅2n−32n−2⋅Note that&amp;nbsp;an+1=an⋅2n+12(n+1)We will show that for all&amp;nbsp;n:14n≤an≤12n+1Having established this, it will follow by comparison with&amp;nbsp;p-series that&amp;nbsp;∑n=1∞anaconverges if and only if&amp;nbsp;a&amp;gt;2We establish the inequalities by induction.Towards showing that&amp;nbsp;an≥12n+1&amp;nbsp;first note that&amp;nbsp;a1≤12⋅1+1Now suppose&amp;nbsp;an≤12n+1. Thenan+12≤[2n+12(n+1)]2⋅12n+1=2n+14(n+1)2=2n+14n2+8n+4≤2n+14n2+8n+1=12n+1So,&amp;nbsp;an≤12n+1&amp;nbsp;for all nTowards showing that&amp;nbsp;an≥14n. Then&amp;nbsp;an+12≥[2n+12(n+1)]2·14n. Now[2n+12(n+1)]2⋅14n−14(n+1)=116n(n+1)2&amp;gt;0thusan+12≥14(n+1)So,&amp;nbsp;an≥14n&amp;nbsp;for all n.

I'm trying to determine the convergence of this

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