Proof of a Bounded Series using the Binomial TheoremDate: 6 Jun 1995 23:53:53 -0400 From: barr Subject: prove (1+1/n)^n is bounded Please help me prove that (1+1/n)^n is bounded from above. I was told to begin by proving: 1+r+r^2+...+r^n = (1-r^(n+1))/(1-r) for any r, [0,r],[r,1) Could I do this simply by multiplying both sides by (1-r)? Next I must use the binominal theorem to show that: (1+1/n)^n is less than or equal to 1+{1+(1/2)+(1/2)^2+...+(1/2)^(n-1)} I can't figure out how to expand the binomial in such a way to show this. Even if I did, how would this prove that (1+1/n)^n is bounded from above? Thank you for your help, Ken Barr P.S. This being our first week of Calculus, I am unfamiliar with differention, integration, etc. Date: 8 Jun 1995 13:03:09 -0400 From: Dr. Ken Subject: Re: prove (1+1/n)^n is bounded Hello there! Could I do this simply by multiplying both sides by (1-r)? Yes, that's what I would do. The left side will then telescope to look just like the right side. ... how would this prove that (1+1/n)^n is bounded from above. Yes, this is kind of tricky. The binomial theorem says that the left side is (1 + 1/n)^n = 1 + [n:1](1/n) + [n:2](1/n)^2 + [n:3](1/n)^3 +...+[n:n](1/n)^n. where the [n:1] and so on means the "choose" symbol, [n:k] = n!/(k!(n-k)!). If n and k are integers, then [n:k] is also an integer. See if you can manipulate the right side of the equation to get the form you're looking for. Hint: how big are the coefficients [n:k]? Once you've done that, look at what you've got. (1+1/n)^n is less than or equal to 1+{1+(1/2)+(1/2)^2+...+(1/2)^(n-1)}, and we can replace the bracketed expression with what we got above and get 1+{(1-.5^n)/(1-.5)} = 1+{1-.5^n/.5} = 1+2(1-.5^n). Now can you show that this thing is bounded? -K Date: 8 Jun 1995 22:15:55 -0400 From: barr Subject: Prove (1+1/n)^n is bounded Thanks for your quick response! (This may be handy in the future!) Our next-year's-calc-teacher explained "n to the n FEPL" to us which helped AND we realized that we were not setting out to show that the limit was e but simply that it HAD a bound. Anyway, thanks again. Have a nice summer, Ken |
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