Euler's Constant: The asymptotic behavior of $\left(\sum\limits_{j=1}^{N} \frac{1}{j}\right) – \log(N)$

I want to show that there exists a constant $\gamma\in\mathbb{R}$ such that

$$\sum_{j=1}^N \frac1{j} = \log(N)+\gamma+O(1/N).$$
I know how to prove that the Euler-Mascheroni constant exists (which I believe $\gamma$ to be), but I am having trouble with the big-$O$ notation and the subsequent bounding. I’ve considered

$$\left|\left(\sum_{j=1}^N \frac1{j}\right) – \log(N)-\gamma\right|\le |K/N|$$ for some $K$, and I was approaching this by trying to show the that the left side of the inequality decays faster, but so far am stuck. Any advice for this type of problem, or analogous ones, would be appreciated.
Thanks!

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Here are some hints. Behind this question stands the absolutely convergent series $\sum\limits_{n\geqslant1} u_n$ defined by
$$u_n=\frac1n-\log\left(\frac{n+1}n\right).$$
You could first try to show that $\sum\limits_{n=1}^{+\infty}u_n$ indeed converges, to a limit $u$ say. Then, due to the cancellations in the logarithms involved in $u_n$, the quantity you are interested in is
$$\left(\sum_{n=1}^N\frac1n\right)-\log(N)=\frac1N+\sum_{n=1}^{N-1}u_n=\frac1N+u-\sum_{n=N}^{+\infty}u_n,$$
hence you could then try to estimate the remaining term $$R_N=\frac1N-\sum\limits_{n=N}^{+\infty}u_n.$$ To do this, an estimate like
$0\leqslant u_n\leqslant\frac1{n(n+1)}$ would be enough, since it implies
$$\frac1N\geqslant R_N\geqslant\frac1N-\sum_{n=N}^{+\infty}\left(\frac1n-\frac1{n+1}\right)=0,$$
showing that $R_n=O\left(\frac1N\right)$ and that $u=\gamma$. Can you prove such an estimate, or a similar one?

You can observe that

$$\sum_{j=1}^n \log( 1+ \frac{1}{j} ) = \log\frac{2}{1} + \log\frac{3}{2}+ \ldots + \log \frac{n+1}{n} = \log{n+1}$$

Then observe that $\sum_{j=1}^n \frac{1}{j} = \log(n+1) + \sum_{j=1}^n \left( \frac{1}{j} – \log( 1+ \frac{1}{j} ) \right)$. The latter sum is actually convergent, since each term is bounded $0 < \frac{1}{j} – \log( 1+ \frac{1}{j} ) < \frac{1}{2j^2}$.

$$\sum_{j=1}^n \left( \frac{1}{j} – \log( 1+ \frac{1}{j} ) \right) = \sum_{j=1}^\infty \left( \frac{1}{j} – \log( 1+ \frac{1}{j} ) \right) – \sum_{j=n+1}^\infty \left( \frac{1}{j} – \log( 1+ \frac{1}{j} ) \right)$$

The sum from positive integers is a negative of the Mascheroni constant. The tail sums is of order $O(n^{-1})$.

Let $[x]$ denote the floor of $x$, and $\{x\}$ the fractional part. Then $$\sum_{k=1}^{N} \frac{1}{k} =1+\int_1^N \frac{[x]}{x^2}dx.$$ Then since $$\int_1^N \frac{[x]}{x^2}dx=\log N -\int_1^N \frac{\{x\}}{x^2}dx,$$ we have $$\sum_{k=1}^{N} \frac{1}{k}= \log N +1-\int_1^N \frac{\{x\}}{x^2}dx.$$

Now, $$\int_1^N \frac{\{x\}}{x^2}dx=\int_1^\infty \frac{\{x\}}{x^2}dx+O\left(\frac{1}{N}\right),$$ so we get the desired result.

By the MVT you have

$$\frac{\ln(k+1)- \ln(k)}{k+1-k}= \frac{1}{c_k} \,.$$

for some $k < c_k < k+1$.

Thus

$$\frac{1}{k+1} < \ln(k+1)- \ln(k) < \frac{1}{k} \,.$$

From here, it is trivial to prove that

$$a_n=1+\frac{1}{2}+..+\frac{1}{n} -\ln (n)$$
is decreasing to $\gamma$

while
$$b_n =1+\frac{1}{2}+..+\frac{1}{n} -\ln (n+1)$$

is increasing to $\gamma$.

It follows immediately that

$$|a_n-\gamma| < |a_n -b_n| \,.$$