Solve Binomial Thm Proof: Prove Increasing & Bounded Sum

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    Binomial Proof
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Homework Help Overview

The problem involves proving a relationship involving the binomial theorem, specifically showing that (1 + 1/n)^n can be expressed as a sum involving factorials and terms that decrease with n. The context includes concepts from calculus and series.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to connect the problem to the binomial theorem but expresses uncertainty in their understanding. They mention needing hints or tips to progress. Some participants suggest using mathematical induction as a potential approach.

Discussion Status

Participants are exploring different methods to tackle the proof, with some suggesting induction as a viable strategy. There is a sense of encouragement among participants, but no explicit consensus on the best approach has been reached yet.

Contextual Notes

The original poster mentions a belief that the sum is increasing and bounded, with an estimated bound of 3, indicating some assumptions about the behavior of the series that are open for discussion.

silvermane
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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!! :blushing: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 :)
 
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Dunno if this will work but have you tried induction?
 
╔(σ_σ)╝ said:
Dunno if this will work but have you tried induction?

Yes induction works.. in fact it is one of the main proof
 
Cool. There you have it silvermane. :-)
 
╔(σ_σ)╝ said:
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 :))
 

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