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I need to find a limit, or approximation for $\sum\limits_{k=1}^{n} (-1)^k {n \choose k} \log(a+bk)$ for, say, an $a,b\in (0,10)$. It is not so important what values $a$ and $b$ have. It would be already helpful for me to find a limit or an approximation for, say, $\sum\limits_{k=1}^{n} (-1)^k {n \choose k} \log(1+2k)$.

This sum arised in some analysis of stochastic processes and I have unfortunately almost not much knowledge in combinatorics and analysis to solve such equations. Thanks for any help.

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    Closely related: http://math.stackexchange.com/questions/64971/proof-sum-limits-k-1n-binomnk-1k-log-k-log-log-n-gamma-fr2012-03-27
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    Thanks, I will look through it. But in fact I know this limit (sorry should have mentioned this). Though I haven't yet tried to understand the proof because it seemed hard and I am not sure if it helps me with my equation. The serie in my question arises from a generalization where the easy case leads to the sum $\sum\limits_{k=1}^n \binom{n}{k}(-1)^k \log k$.2012-03-27
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    @Coopi: because your account is not registered, when you change computers or IP addresses the system sometimes "forgets" who you are. Please consider registering your account: this will guarantee you the ability to edit and comment on your own questions.2012-04-04

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