# Check the convergence of $\sum_{n=1}^{\infty} ( e - (1+ \frac{1}{n})^{n})$ [duplicate]

Check the convergence of $$\sum_{n=1}^{\infty} ( e - (1+ \frac{1}{n})^{n})$$

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

So, divergence test fails. I couldn't find any suitable test to check the convergence of this series.

Any suggestion $$?$$

• There are some more answers here – Zacharias Zarowski Dec 4 '19 at 14:27
• – Martin R Dec 4 '19 at 14:28

$$\left(1+ \frac{1}{n}\right)^{n}=e^{n\log \left(1+ \frac{1}{n}\right)}=e^{1-\frac1{2n}+O\left(\frac1{n^2}\right)}=e\left(1-\frac1{2n}+O\left(\frac1{n^2}\right)\right)$$
$$e - \left(1+ \frac{1}{n}\right)^{n}=\frac e{2n}+O\left(\frac1{n^2}\right)$$