How Does Binomial Expansion Help Define e Using Limits?

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The discussion focuses on using binomial expansion to demonstrate that e can be defined as the limit of (1 + 1/n)^n as n approaches infinity. The user is attempting to express this limit in terms of a series involving factorials, specifically aiming to show that it equals the sum of 1/k! for k from 0 to infinity. They seek guidance on manipulating the expressions to eliminate n and simplify the series. The conversation also touches on the Stirling formula as a potential tool for expanding factorials in this context. Overall, the user is looking for foundational resources to better understand these mathematical concepts.
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Hi!

I'm currently taking a fairly early stats course, and I'm having a bit of a hangup learning exactly how to use "moments" properly. My general solution whenever I run into problems internalizing things is to do a bunch of easy problems, and to show it from the ground up.

This is my first post here, so I'm not certain how to use the math notationy stuff.

My issue here is fairly simple - I want to show that e = lim n-> inf (1+1/n)^n = 1/0!+1/1!+1/2!+...

So I start with my n-> inf (1+1/n)^n and do binomial expansion.

So e = lim n-> Infinity Sum[(n choose k)*1/n^k, {k, 0, Infinity}].

From here, we can go to Sum[n!/(n-k)!k! * 1/n^k, {k,0,Infinity}]

Next... Okay, next we can change the numerator to (n-1)! and the denominator of the right to n^(k-1), but I have no idea what I can do next, considering I'm trying to get to Sum[1/k!,{k,0,Infinity}]. How do I remove the n's?

Thank you very much, and apologies for the mess.
 
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Do you know the Stirling formula? You can use it to expand some factorials.
 
Nope, wikipedia doesn't seem to start at the ground for it, either. :( I'll youtube it!
 
What is wrong with the formulas here?
$$n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$$
The ratio of both goes to 1 in the limit n->infinity
 
Perfect, that is exactly the sort of thing I was looking for.

Thanks! I think I need to learn more calculus.
 

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