Solve Binomial Thm Proof: Prove Increasing & Bounded Sum

[silvermane]

Homework Statement

Prove that (1 + 1/n)^n = 1 + \sum1/m!(1 - 1/n)(1-2/n)...(1-(m-1)/n)
where our sum is from m=1 to n.

2. Attempt:
I recognize the binomial theorem here, but I'm having a mental block in how to figure this out. I should know how to do this, I think I just need a little help getting the neurons firing...
Any hints or tips are greatly appreciated!! Also, for my own enrichment, I came to the conclusion that this is increasing, and bounded. I think that the nth term is less than it's bound too, which is estimated at 3. If anyone can help with this, it would be great for further understanding with sums. I may have thought too much into this, but let me know what your thoughts are too on the matter. I'm just trying to brush up on my calculus skills :)

 

[╔(σ_σ)╝]

Dunno if this will work but have you tried induction?

 

[icystrike]

Dunno if this will work but have you tried induction?

Yes induction works.. in fact it is one of the main proof

 

[╔(σ_σ)╝]

Cool. There you have it silvermane. :-)

 

[silvermane]

Cool. There you have it silvermane. :-)

lol awesome!
I just was thinking that it was the most reasonable way to tackle the problem.

Thanks again!
I'll post if I have any other questions