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.