Sequence question. Need help

1. Apr 22, 2007

sutupidmath

sequence question. Need help!!

i would like to know where could i find the proof that the sequence

a_n=(1+1/n)^n bounded (upper bounded) by 4.

or in general that this sequence is a convergent one??

i know the proof by expanding it using binominal formula(Newton formula), but i am looking for another proof, by using some other helping sequence, and than to tell that this sequence a_n=(1+1/n)^n is smaller than every term of the helpin sequence????

any help would be appreciated.

2. Apr 22, 2007

mathman

Have you looked at the power series for e?

3. Apr 23, 2007

sutupidmath

do u mean expressing e using taylor formula??

4. Apr 23, 2007

Gib Z

I Think thats what he meant, but heres a definition of e that will help :)

$$e=\lim_{n\to\infty} (1+\frac{1}{n})^n$$.
So as n goes to infinity, a_n goes to e, which is less than 4.

However, you need to know that a_n < 4 for ANY n. You know a_1=2.

So to make sure for any positive integer n, 2<a_n<4, we show that that a_n is an monotonically increasing function for positive n.

Last edited: Apr 23, 2007
5. Apr 23, 2007

sutupidmath

Yes, i do know this. But what i am looking for is a proof(another proof, couse i already know two of them) using another sequence that will look something like this

b_n=(1+1/(n^2-1))^n

and than to show that for every n, a_n, is less than b_n, for every n.(a_n<b_n)

because i know that 2<e<3<4 .

because i need to prove it this whay(using another helping sequence which charasteristics we know).

thnx anywhay

6. Apr 23, 2007

Gib Z

Well I'm not sure about the form of your b_n, but heres another b_n thats larger term by term and is bounded by 4 as well.

$$e^x = \lim_{n\to\infty} (1+\frac{x}{n})^n$$
So a b_n that is larger term by term and bounded we be say..b_n = (1 + 1.002/n)^n? Any value of or less than ln 4 will do in place of 1.002.

7. Apr 23, 2007

mathman

e=1+1/1!+1/2!+1/3!+.... which dominates the binomial expansion of (1+1/n)n

8. Apr 24, 2007

sutupidmath

How does this help, to prove what i am looking for???