Series$\sum_{n=1}^\infty\big((1+1/n)^n-e\big)$

Take the negative of that series to make the terms positive and limit compare it with $\frac{1}{n}$

$$\lim_{n\to\infty} \frac{e-\left(1+\frac{1}{n}\right)^n}{\frac{1}{n}} \to \frac{0}{0}$$

$$\implies \lim_{n\to\infty} \frac{-\left(1+\frac{1}{n}\right)^n\Biggr(\log\left(1+\frac{1}{n}\right)-\frac{1}{n+1}\Biggr)}{-\frac{1}{n^2}}$$

which we get from L'Hopital and logarithmic differentiation. If this limit exists, we can decompose it into a product. The limit on the left is $e$, so let's evaluate

$$\lim_{n\to\infty} \frac{\log\left(1+\frac{1}{n}\right)-\frac{1}{n+1}}{\frac{1}{n^2}}\to \frac{0}{0}$$

$$\implies \lim_{n\to\infty} \frac{\frac{1}{1+\frac{1}{n}}\cdot\left(-\frac{1}{n^2}\right)+\frac{1}{(n+1)^2}}{-\frac{2}{n^3}} = \lim_{n\to\infty}\frac{1}{2}\frac{n^2}{(n+1)^2} = \frac{1}{2}$$

This means the original limit equals $\frac{e}{2}$, so the series behaves as $\frac{1}{n}$, i.e. it diverges by the limit comparison test.

As an answer hasn't been accepted yet, I assume that there's something left to be desired. So I'll take a different stab at the problem that only involves calculus.

We first observe, as in the comments, that $$\Bigl(1+\frac{1}{n}\Bigr)^n-e=e^{n\ln(1+1/n)}-e\,.$$

Elementary calculus arguments can show that the function $x\ln(1+1/x)$ is monotone increasing on $(0,\infty)$ and that its range has $1$ as a supremum and has $0$ as an infimum.

We now invoke the Mean Value Theorem to conclude that for every $n\in\mathbb{N}$ there is a real $\xi_n$ between $1$ and $n\ln(1+1/n)$ such that $$e-\Bigl(1+\frac{1}{n}\Bigr)^n=e^{\xi_n}\bigl(1-n\ln(1+1/n)\bigr)\,.$$

From what we know about $x\ln(1+1/x)$, we can conclude that $$e^0\bigl(1-n\ln(1+1/n)\bigr)\leq e-\Bigl(1+\frac{1}{n}\Bigr)^n\leq e^1\bigl(1-n\ln(1+1/n)\bigr)\,.$$

Now we use the inequalities \begin{equation}\frac{1}{2(1+x)}\leq 1-x\ln(1+1/x)\leq\frac{1}{x+1}\end{equation} which is true for all positive $x$ to conclude $$\frac{1}{2(1+n)}\leq e-\Bigl(1+\frac{1}{n}\Bigr)^n\leq\frac{e}{1+n}$$ which will carry the rest of the argument.

The real crux to this argument are the inequalities \begin{equation}\frac{1}{2(1+x)}\leq 1-x\ln(1+1/x)\leq\frac{1}{x+1}\,.\end{equation}

The right-hand inequality is equivalent to the more familiar inequality $$\ln(1+x)\leq x$$ which I can derive if needed.

The left-hand inequality is equivalent to the slightly stronger inequality $$\ln(1+x)\leq x-\frac{x^2}{2(1+x)}$$ which is however established in very much the same way as the weaker inequality.