I am trying to show that for every $\epsilon0$, there exists an $N\in\mathbb{N}$ such that$$\left|\left(\frac{x}{n}+1\right)^n-e^x\right|

Question

I am trying to show that for every $\epsilon>0$, there exists an $N\in\mathbb{N}$ such that$$\left|\left(\frac{x}{n}+1\right)^n-e^x\right|<\epsilon$$whenever $n\geq N$. How can I go about doing this?

This a problem about uniform convergence.

anonymous · Answer

By the way, $n\in\mathbb{N}$ and $x\in[-A,A]$.

anonymous · Answer

@satellite73 @Zarkon @JamesJ @Vernard_Mercader_PhD @brinethery

jamesj · Answer

The careful proof of this is not achieved in a couple of lines.

One book that works through this carefully is Spivak's "Calculus"

Spivak is my favorite introduction to analysis and I recommend you pick up a copy, even if it's from your library.

anonymous · Answer

I'm so stuuuupid!  That was from 6 years ago, I can't even remember what I had for breakfast lol.