The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications

ddaeeric 2021-09-30 Answered

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $ limna2nan<1/2 then an converges, while if limna2n+1an>1/2 then an  diverges. Let an=nlnn(lnn)n. Show that a2nan0 as n

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Jayden-James Duffy
Answered 2021-10-01 Author has 91 answers

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The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence.

The test states that if

 limna2nan<1/2

then an converges,while if

limna2n+1an>1/2,

then an  diverges. Let an=11+x22+xnn+x1n=(n1)!(1+x)(2+x)(n+x).

Show that a2n/anex/2/2  .

For which x > 0 does the generalized ratio test imply convergence of n=1an?

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