How to Prove Euler's Number Converges?

[jetplan]

Hi All,

How do we go about showing the euler's number e converges ?
Recall that

e =(1+1/n)^n as n ->infinity

Some place prove this by showing the sequence is bounded above by 3 and is monotonic increasing, thus a limit exist.

But I forgot how exactly the proof looks like.


Thanks
J

 

[hamster143]

Here's one way.

Write down (1+1/n)^n using the binomial formula.

You'll see that it equals to

1 + 1 + (1/2!)(1-1/n) + (1/3!) (1-1/n) (1-2/n) + ... + (1/(n-1)!) (1-1/n) (1-2/n) .. (1-(n-1)/n)

From here it's not hard to see that it's monotonic increasing, and it's bounded above by 1 + 1 + 1/2! + 1/3! + ... 1/n!.

In turn, it's not hard to see that the series $\sum 1/n!$ converges, because $1/n! < 2^{-n}$ for all n>3.

 

[lurflurf]

Use the fact that
geometric mean<=arithmetic mean
consider n+1 numbers where 1 is one of the numbers and the other n are 1+1/n

 

[hamster143]

Use the fact that
geometric mean<=arithmetic mean
consider n+1 numbers where 1 is one of the numbers and the other n are 1+1/n

That only proves that it's monotonic.

 

[jetplan]

Thanks for the neat proof.

one comment:

since we know that $\sum 1/n!$ converges, we know e converges because
e = 1 + $\sum 1/n!$ where n->infinity

 

[hamster143]

Thanks for the neat proof.

one comment:

since we know that $\sum 1/n!$ converges, we know e converges because
e = 1 + $\sum 1/n!$ where n->infinity

but first you'd have to prove that the original series converges and that its limit is equal to 1 + $\sum_{n=1}^{\infty} 1/n!$ .

 

[lurflurf]

That only proves that it's monotonic.

2<(1+1/n)^n<(1+1/n)^(n+1)<4

By Monotone convergence theorem bounded monotonic series converge.

 

[hamster143]

(1+1/n)^(n+1)<4

I don't see it.

 

[lurflurf]

^
(1+1/n)^(n+1) is decreasing thus
(1+1/n)^(n+1)<(1+1/1)^(1+1)=4

 

[HallsofIvy]

You repeated use of "e is convergant" is confusing. Numbers do not converge. It makes sense to ask how to show that the limit $\lim_{n\to\infty} (1+ 1/n)^n$ converges or how to show that $\sum_{n=0}^\infty 1/n!$ converges.