Value of $\psi\left(\frac{1}{2}\right)$ I apologise if this is a dumb question, but I have trouble deriving $\displaystyle\psi\left(\frac{1}{2}\right)=-\gamma-2\ln{2}$. I have tried the following.
\begin{align}
\psi\left(\frac{1}{2}\right)
&=\lim_{N\to\infty}\left[-\gamma-2+\sum^N_{k=1}\left(\frac{1}{k}-\frac{1}{k+1/2}\right)\right]\\
&=\lim_{N\to\infty}\left[-\gamma-1+\sum^{\lfloor \frac{N-1}{2} \rfloor}_{k=1}\frac{1}{2k}+\sum^{\lfloor \frac{N-1}{2} \rfloor}_{k=1}\frac{1}{2k+1}-2\sum^N_{k=1}\frac{1}{2k+1}\right]\\
&=\lim_{N\to\infty}\left[-\gamma-1+\sum^{\lfloor \frac{N-1}{2} \rfloor}_{k=1}\left(\frac{1}{2k}-\frac{1}{2k+1}\right)-2\sum^N_{k=1+\lfloor \frac{N-1}{2} \rfloor}\frac{1}{2k+1}\right]\\
&=-\gamma-\ln{2}-\lim_{N\to\infty}\sum^N_{k=1+\lfloor \frac{N-1}{2} \rfloor}\frac{2}{2k+1}
\end{align}
I have trouble showing that the last sum $\to \ln{2}$ as $N \to \infty$, and would like to seek your assistance in evaluating the limit. Thank you in advance.

One could evaluate the limit as follows. Set $N=2 n$:
$$\lim_{N\to \infty} \sum_{k=1+\lfloor{\frac{N-1}{2}}\rfloor}^{N} \frac{2}{2k+1}=\lim_{n\to \infty} \sum_{k=n}^{2n} \frac{2}{2k+1}=\lim_{n\to \infty} 2\cdot\left(\sum_{k=0}^{2n} \frac{1}{2k+1}-\sum_{k=0}^{n-1} \frac{1}{2k+1}\right)=2\cdot\lim_{n\to \infty} \left(\left(\sum_{k=1}^{4n+1} \frac{1}{k}-\sum_{k=1}^{2n} \frac{1}{2k}\right)-\left(\sum_{k=1}^{2n-1} \frac{1}{k}-\sum_{k=1}^{n-1} \frac{1}{2k}\right)\right)=2\cdot\lim_{n\to \infty}{H_{4n+1}-\frac{1}{2}H_{2n}-H_{2n-1}+\frac{1}{2}H_{n-1}}=2\cdot\lim_{n\to \infty}{(H_{4n+1}-\ln(4n+1))+\ln(4n+1)-\frac{1}{2}(H_{2n}-\ln(2n))-\frac{1}{2}\ln(2n)-(H_{2n-1}-\ln(2n-1))-\ln(2n-1)+\frac{1}{2}(H_{n-1}-\ln(n-1))+\frac{1}{2}\ln(n-1)}=2\cdot\lim_{n\to \infty}{(H_{4n+1}-\ln(4n+1))-\frac{1}{2}(H_{2n}-\ln(2n))-(H_{2n-1}-\ln(2n-1))+\frac{1}{2}(H_{n-1}-\ln(n-1))}+2\cdot\lim_{n\to \infty}{\ln(4n+1)-\frac{1}{2}\ln(2n)-\ln(2n-1)+\frac{1}{2}\ln(n-1)}=2\cdot(\gamma-\frac{1}{2}\gamma-\gamma+\frac{1}{2}\gamma)+2\cdot\lim_{n\to \infty}{\ln(\frac{4n+1}{2n-1})-\frac{1}{2}\ln(\frac{2n}{n-1})}=2\cdot(\ln(2)-\frac{1}{2}\ln(2))=\ln(2)$$
Here I used that:
$$\lim_{n\to \infty}H_n-\ln(n)=\gamma$$
Where $\gamma$ is Euler's constant.
Note that there is a faster way to evaluate $\psi\left(\frac{1}{2}\right)$:
$$\psi\left(\frac{1}{2}\right)=-\gamma-2+\sum_{k=1}^{\infty} \left(\frac{1}{k}-\frac{1}{k+\frac{1}{2}}\right)=-\gamma+\sum_{k=1}^{\infty} \left(\frac{1}{k}-\frac{1}{k-\frac{1}{2}}\right)=-\gamma+\sum_{k=1}^{\infty} \left(\frac{2}{2k}-\frac{2}{2k-1}\right)=-\gamma-2\cdot\sum_{k=1}^{\infty} \left(\frac{1}{2k-1}-\frac{1}{2k}\right)=-\gamma-2\cdot\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}=-\gamma-2\ln(2)$$
I hope, this is helpful and I apologize for all mistakes in my English, I'm not a native speaker. :)
