I'm trying to prove the existence of $\lim_{n\rightarrow\infty}({\sum_{k=1}^{n}{1/k}-\log(n+1)})$, I have used the Integral Test Theorem applied to $f(x)=1/x(x+1)$ but I got nowhere unfortunately.
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1Does this answer your question? Why is $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} \approx \ln(n) + \gamma$? – nejimban Oct 04 '21 at 14:56
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Check https://en.wikipedia.org/wiki/Euler%27s_constant – Maksim Oct 04 '21 at 15:02