Deriving the constant e using a sequence limit

[Fibo's Rabbit]

Homework Statement

Why does lim( (1+(1/n))^n ) = e?

Homework Equations

If a_n convergent to a, and b_n converges to b, then (a_n * b_n) converges to (a * b)

The Attempt at a Solution

The lim(1 + (1/n)) = 1.

If you multiply (1 + (1/n)) by itself n-times, you get the equation (1 + (1/n))^n, so according to the statement under Relevant equations above, shouldn't the answer be 1^n, or just 1?

Obviously it is e, b/c when you plug (1 + (1/n))^n into your calculator for larger values of n, you get closer and closer to e.

 

[tiny-tim]

Hi Fibo's Rabbit!

(try using the X² and X₂ buttons just above the Reply box )

If a_n convergent to a, and b_n converges to b, then (a_n * b_n) converges to (a * b)

ah, but that only applies for the product of a fixed number of series …

you're trying to apply it to an infinitely increasing number of series ​

 

[Fibo's Rabbit]

Ah, and it all makes a lot more sense now.