<s>[INST] Find $\lim\limits_{n\to\infty} 1 + 1/2 + 1/3+\cdots+1/2n- \ln n$ I know the proof of  $$\lim_{n\to\infty}  1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \ln n = \gamma$$ but how do I use that proof to find the value of the limit above.
 [/INST] \begin{align}
& 1 + \frac 12 + \frac 13 + \cdots + \frac 1 {2n} -\ln n \\[10pt]
= {} & \Big(\  \underbrace{1 + \frac 12 + \frac 13 + \cdots + \frac 1 {2n} -\ln(2n)}_{\text{This part approaches $\gamma$.}}\  \Big) + \ln 2.
\end{align}
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