The following advanced exercise use a generalized ratio test to determine convergence of some series

alesterp

alesterp

Answered question

2021-05-05

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 $$ lim{n}a{2n}an<12 then anconverges,while if lim{n}a{2n+1}an>12, then an diverges.

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

Show that a2nanex/22 . For which x > 0 does the generalized ratio test imply convergence of n=1an?

Answer & Explanation

Malena

Malena

Skilled2021-05-06Added 83 answers

a2nanex22
No, he ratio test does NOT imply the convergence for n=1an.

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