Understanding the Limit of (1+1/n)^n as n Goes to Infinity

[moo5003]

I'm studying for the GRE that's coming up in a week or two and I came across a problem where the answer given in the book does not make sense to me and I was wondering of someone here could explain it to me.

Question:

Lim as n goes to infinity of X_(n+1) / X_n

Where X_n = n^n / n!

Answer:

So I started by simplifying the expression down to:

Lim as n goes to infinity of (1 + 1/n)^n

The book informs me and by some proofs online that this tends toward e. However I was hoping someone could explain this to me because from my point of view it should just hit 1.

Since 1/n -> 0, 1+0 = 1, and 1^n is 1 for any arbitrarly high power.

 

[mathman]

Use the binomial theorem on (1 + 1/n)^n and see what happens to each term as n becomes infinite. You will end up with the power series for e.

 

[moo5003]

So, using binomial expansion we get:

Lim_n^Inf of
Series m=0 to n

(1)^(n-m) * (1/n)^m * nCm

Note: nCm = n!/[m!(n-m)!]

I'm a little unsure how to simplify this. Obviously from your response I'm looking to simplify this to 1/m!.

Assuming I can pass the limit through the series (unsure of this) we get:

Series m=0 to Infinity
Lim_n^Inf of
(1/n)^m * n!/[m!(n-m)!]

What next?

 

[moo5003]

Bump* Just wanted to see if anyone could clarify this question for me.

 

[Mark44]

I'm studying for the GRE that's coming up in a week or two and I came across a problem where the answer given in the book does not make sense to me and I was wondering of someone here could explain it to me.

Question:

Lim as n goes to infinity of X_(n+1) / X_n

Where X_n = n^n / n!

Answer:

So I started by simplifying the expression down to:

Lim as n goes to infinity of (1 + 1/n)^n

The book informs me and by some proofs online that this tends toward e. However I was hoping someone could explain this to me because from my point of view it should just hit 1.

Since 1/n -> 0, 1+0 = 1, and 1^n is 1 for any arbitrarly high power.

I thought it would be worthwhile to jump in here and point out that your analysis is faulty in the line just above. Yes, it's true that 1/n --> 0 as n gets large, and 1^n is 1 for any arbitrarily large finite power, but this is not the way limits work. The limit process applies to the whole expression, not just a bit here and another bit there later on.

The point is that, although 1/n --> 0 in the limit, for any finite value n, 1/n is not zero, so for the same value of n, (1 + 1/n) ^ n is not 1^n.

So while the base is getting closer to 1, the exponent is getting larger and larger. This type of limit is one of several that are called indeterminate forms. I'm reasonably sure you can do a search on wikipedia to find a page with more information. They are called indeterminate because you can't determine at a glance what their limits will be.