How do you do this transformation of two Euler's number formula?

Asked May 24 '22 at 21:37

Active May 24 '22 at 22:10

Viewed 31 times

I found a simple programming problem that asks how to get Euler's number by iteration. I do not talk about this problem here, but when I researched the definition of Euler's number (well, I think I am already learned them), I found two formulas.

One has the definition of limit, and another is defined by partial sum. So, I raise the question. How can these different formulas transform into each other? Is it even possible?

If it's possible,

How to prove/transform left to right?

$$\lim_{n \to \infty}(1+{1\over{n}})^n = \sum_{n=0}^\infty {1 \over n!} $$

off-topic note

I think I could solve the problem by using partial sum definition. the program like this

double a[2]={1,1}
for(int i=1;i>1;i++){
   a[0]*= 1.0/i
   a[1]+= a[0]
   if (a[0]<pow(0.1,15){break;}
}
printf("e=%f",a[1])

edited May 24 '22 at 22:10

asked May 24 '22 at 21:37

nobiu

Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun May 24 '22 at 21:42
Yes, Thank you so much! I couldn't figure out how to search for this problem. – nobiu May 24 '22 at 21:44
@Shaun Sorry, I will edit this now – nobiu May 24 '22 at 21:45
Also see Tannery's theorem on Wikipedia. – bjorn93 May 24 '22 at 21:51
@bjorn93 I am happy to know this theorem! Thank you! – nobiu May 24 '22 at 22:05
Also, it is $1+1/n$, not $1+1/n!$. – marty cohen May 24 '22 at 22:08
@marty cohen Oh, yes, I have corrected it! Thank you so much!! – nobiu May 24 '22 at 22:09

How do you do this transformation of two Euler's number formula?

off-topic note

0 Answers0