How do I show a series converges or diverges?

[tardy hamlet]

Hi. I thought of the infinite series $\sum_{n=1}^{\infty}( (1 + \frac {1}{n})^{n} - e)$ and wondered if the series converges or diverges. As you take higher n, the series grows, and “at infinity” it is 0. How do I proceed in showing it?

[lament pier]

CJ_:)

[tardy hamlet]

<@&286206848099549185>

@candid halo

[strange gust]

Sage says it's divergent, btw

[teal lava]

e-(1+1/x)^x>1/x when x>3 and if we rerange terms we get e>1/x+(1+1/x)^x after that i got stuck 🍮

[cerulean flame]

It is necessary for the general term to go to zero but it is not sufficient

What you can do is write
$(1+\frac{1}{n})^n
= exp(n ln(1+ \frac{1}{n}))$

[lament pier]

Daddy_314

[cerulean flame]

Then expand ln(1+x) when x = 1/n approaches 0

[tardy hamlet]

,close

[austere wadi]

.close