Can anyone help me with the proof that $a_n=\sum_{i=0}^{n}\frac{1}{n\choose i}$ converges.Now involving that I have two questions.Firstly is there a closed formula for $a_n$?And secondly is the general series convergent? $a_{n,s}=\sum_{i=0}^{n}\frac{1}{{n\choose i}^s}$.
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Depends on $s$. Certainly not for, say, $s=-1$. – Gerry Myerson Nov 24 '13 at 05:38
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See this answer http://math.stackexchange.com/questions/151441/calculate-sums-of-inverses-of-binomial-coefficients – emi Nov 24 '13 at 07:27
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The general series {$a_{n,s}$} is not convergent, at least not for all $s$; for $s=1$, it will not, because :
$a_n=\sum_{i=0}^{n}\frac{1}{n\choose i}$ will contain ${n\choose 0}={n\choose 1}=1$ , so that $a_n >2 $ for all $n$
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