Can I move the limit inside (1+1/n)^n?

[jeffreylze]

Homework Statement

http://img132.imageshack.us/img132/6663/captureqqi.jpg

Homework Equations

The Attempt at a Solution

I came up with this answer and i don't know how to continue:

(1+1/n)^n

I understand I have to put in limit n -> infinty. But can i put the limit inside the bracket? So i will get an answer of 1 ?

So ultimately, the ratio test is inconclusive?

 

[dx]

(1 + (1/n))ⁿ → e (≈ 2.718) as n → ∞.

 

[jeffreylze]

How did you get that? Can you bring me through the steps?

 

[dx]

It's a standard limit. Use the binomial theorem to expand (1 + (1/n))ⁿ, and then take the limit n → ∞. You should get

1 + 1 + 1/2! + 1/3! + ... = e