Solve Binomial Thm Proof: Prove Increasing & Bounded Sum

In summary, the conversation revolved around proving the equation (1 + 1/n)^n = 1 + \sum1/m!(1 - 1/n)(1-2/n)...(1-(m-1)/n), where the sum is from m=1 to n. The individual was having trouble solving it using the binomial theorem and asked for hints or tips. Another person suggested using induction, which was confirmed to be a valid approach. The conversation ended with the individual expressing gratitude and mentioning that they may have further questions.
  • #1
silvermane
Gold Member
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0

Homework Statement


Prove that (1 + 1/n)^n = 1 + [tex]\sum1/m!(1 - 1/n)(1-2/n)...(1-(m-1)/n)[/tex]
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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  • #2
Dunno if this will work but have you tried induction?
 
  • #3
╔(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
 
  • #4
Cool. There you have it silvermane. :-)
 
  • #5
╔(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 :))
 

1. What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand a binomial expression raised to a positive integer power. It can be written as (a + b)^n = Σ(n,k=0) (n choose k) * a^(n-k) * b^k, where n is the power, a and b are the binomial terms, and (n choose k) is the binomial coefficient.

2. How is the binomial theorem used to solve proofs?

The binomial theorem can be used to prove various mathematical statements involving binomial expressions, such as the increasing and bounded sum. By expanding the binomial expression using the theorem, we can manipulate and simplify the resulting terms to prove the desired statement.

3. What does it mean for a sum to be increasing and bounded?

An increasing sum means that as the number of terms in the sum increases, the overall value of the sum also increases. A bounded sum means that the sum has an upper limit or maximum value, beyond which it cannot increase.

4. How do you prove that a sum is both increasing and bounded?

To prove that a sum is both increasing and bounded, we can use the binomial theorem to expand and manipulate the terms in the sum. By showing that the resulting terms are in a specific form, we can prove that the sum is both increasing and bounded.

5. Can the binomial theorem be used to prove other statements besides increasing and bounded sums?

Yes, the binomial theorem can be used to prove various mathematical statements involving binomial expressions, such as the binomial theorem itself, Pascal's triangle, and the binomial distribution. It can also be applied in other fields such as physics and engineering.

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