(adsbygoogle = window.adsbygoogle || []).push({}); The proof of Raabe's test

1. The problem statement, all variables and given/known data

I was asked to prove Raabes test in the form that if

[tex]|a_{n+1}/a_{n}|<1-A/n[/tex]

for some A>1 and for n large enough, then the series converges absolutely.

After struggling for days (I work at a factory doing repetitive work so I can easily let my body work while I think about math ), I capitulated and peeked at the hint at the end of the book. It says, let

[tex]P_n=\prod_{k=1}^{n}\left(1-\frac{A}{k}\right)[/tex]

and show that

[tex]\ln(P_n)=-A\ln(n)+O(1)[/tex].

While I understand perfectly how this equality is the kernel of the proof, I can't seem to be able to demonstrate its truth. My best shot is...

[tex]\ln(P_n)=\sum_{k=1}^n\ln\left(1-\frac{A}{k}\right)\leq \sum_{k=1}^n\left(1-\frac{A}{k}\right)=n-A\sum_{k=1}^n\frac{1}{k}=n-A\ln(n)+O(1)[/tex]

The last equality is because the sequence [tex]\gamma_n=\sum_{k=1}^n\frac{1}{k}-\ln(n)[/tex] is decreasing and bounded below by 0 (and converges to Euler's cst [itex]\gamma[/itex]).

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# The proof of Rabbe's test

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