find the limit of this sequence

[trickslapper]

X[sub:2e0xsn4g]n[/sub:2e0xsn4g]=(1+(1/n))[sup:2e0xsn4g]n[/sup:2e0xsn4g], where n is a natural number.

I know the answer is e, because i checked it in the calculator but i have to show this is true by:

1. establishing monotonicity (prof said to use binomial formula for this)
2. and establishing an upperbound (if the sequence is increasing, which it is)

*Im guessing once we expand using the binomial formula, a bunch of things will go to zero (as n gets larger and larger) leaving 2.7.... as my answer?

 

[galactus]

See here:

http://galileo.phys.virginia.edu/classe ... nction.htm

 

[pka]

Suppose that \(\displaystyle n\ge 2\).
Then \(\displaystyle \left( {1 + \frac{1}{n}} \right)^n \left( {1 - \frac{1}{n}} \right)^n = \left( {1 - \frac{1}{{n^2 }}} \right)^n \geqslant \left( {1 - \frac{1}{n}} \right)\).
Divide both sides.
\(\displaystyle \left( {1 + \frac{1}{n}} \right)^n \geqslant \left( {1 - \frac{1}{n}} \right)^{1 - n} = \left( {1 + \frac{1}{{n - 1}}} \right)^{n - 1}\)
That shows it is increasing.

Can you do bounded?