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Posers and Puzzles

Posers and Puzzles

  1. Donation !~TONY~!
    1...c5!
    26 Oct '04 02:24
    I am studying for a quiz tomorrow in Calc II and ran across I problem that I know we did in class but I don't know how to do it and didn't copy it down. (idiot, I know). The question is : Find whether the sequence (N + 1/ N) ^ N from N= 1 to infinity converges or diverges, and if it converges, where? The answer is e, but I didn't some mumbo jumbo that didn't work, and I don't even know if it's mathematically legal. Anyone know how to do this?
  2. Standard member telerion
    True X X Xian
    26 Oct '04 04:50
    Originally posted by !~TONY~!
    I am studying for a quiz tomorrow in Calc II and ran across I problem that I know we did in class but I don't know how to do it and didn't copy it down. (idiot, I know). The question is : Find whether the sequence (N + 1/ N) ^ N from N= 1 to infinity converges or diverges, and if it converges, where? The answer is e, but I didn't some mumbo jumbo that didn't work, and I don't even know if it's mathematically legal. Anyone know how to do this?
    Well let's see. Just so that I'm sure. Is it [(n+1)/n]^n or is it [n+(1/n)]^n?

    The first one just goes to 1. Since it can be rewritten

    [1+(1/n)]^n which as n-> infinite goes to 1^n = 1.

    So I don't suppose that's the version of it.

    The second one clearly diverges. Again in the limit, 1/n -> 0, and so the sequence -> infinite^infinite .

    You're sure the answer is e?

    Maybe the question was a little different?

    Well, that's my two cents. I'll differ to one of the mathematicians on the site if I'm missing something.
  3. Subscriber AThousandYoung
    It's about respect
    26 Oct '04 08:14 / 1 edit
    http://mathworld.wolfram.com/e.html

    e is defined as lim(n=>infinity) (1+(1/n))^n = ((n+1)/n)^n

    There's all kinds of stuff about e there.

    I'm trying to find the answer to your question now.
  4. Subscriber AThousandYoung
    It's about respect
    26 Oct '04 08:29
    [1+(1/n)]^n which as n-> infinite goes to 1^n = 1.

    This is flawed, because even as 1/n gets very very small, the power of the sum of 1+(1/n) gets really, really huge.
  5. Subscriber AThousandYoung
    It's about respect
    26 Oct '04 08:41
    This looks like it might help you.

    http://mathforum.org/library/drmath/view/51954.html

    Use the Binomial Theorem to expand the limit.

    (1 + 1/n)^n = 1 + 1 + (1 - 1/n)/2! + (1 - 1/n)(1 - 2/n)/3! + ... + (1 - 1/n)(1 - 2/n)(...)(1 - (n-1)/n)/n!.

    The last term clearly approaches zero, since n! approaches infinity and none of the terms in the numerator approach infinity. In fact, (1 - (n-1)/n) approaches zero by itself (1-1). So, the series converges.

    I don't know how to prove the answer is e though. Maybe you can wade through those websites and figure it out.
  6. Standard member royalchicken
    CHAOS GHOST!!!
    26 Oct '04 11:24 / 1 edit
    If it's (n+1/n)^n then it diverges. If it's (1+1/n)^n, then it converges to e as follows:

    (1+1/n)^n = 1+(n/n)+n(n-1)*n^-2/2 + n(n-1)(n-2)*n^-3/6... +n^-n

    Letting n --> infinity and canceling ns:

    = 1 + 1 + 1/2 + 1/6 + ...

    = e.

    EDIT The last bit is just the definition of e.
  7. Donation !~TONY~!
    1...c5!
    26 Oct '04 16:41
    I figured it out today before the quiz. Good thing there was nothing of the sort on the test. Instead he unleashed this one. A ball is dropped from 6 feet above the ground, with each subsequent bounce being (3/4) that of the next. What is the total vertical distance traveled by the ball. I got it, but it took me about 15 minutes!
  8. Standard member telerion
    True X X Xian
    26 Oct '04 17:08 / 1 edit
    Originally posted by !~TONY~!
    I figured it out today before the quiz. Good thing there was nothing of the sort on the test. Instead he unleashed this one. A ball is dropped from 6 feet above the ground, with each subsequent bounce being (3/4) that of the next. What is ...[text shortened]... traveled by the ball. I got it, but it took me about 15 minutes!
    Oh yeah I remember those. Nice little application of a geometric series. I think I had that exact problem back win I took calculus.

    Thank you all for the correction on e.

  9. Donation !~TONY~!
    1...c5!
    26 Oct '04 17:19
    Originally posted by telerion
    Oh yeah I remember those. Nice little application of a geometric series. I think I had that exact problem back win I took calculus.

    Thank you all for the correction on e.

    Yeah, it took me quite a bit to figure out it was a geometric series. I was messing around with telescopic series' for a about 5 minutes before that...hehehehehe....
  10. 27 Oct '04 10:57
    Originally posted by royalchicken
    If it's (n+1/n)^n then it diverges. If it's (1+1/n)^n, then it converges to e as follows:

    (1+1/n)^n = 1+(n/n)+n(n-1)*n^-2/2 + n(n-1)(n-2)*n^-3/6... +n^-n

    Letting n --> infinity and canceling ns:

    = 1 + 1 + 1/2 + 1/6 + ...

    = e.

    EDIT The last bit is just the definition of e.
    e = 2.7 1828 1828 459045 2353 6028 7549 ......
  11. Standard member royalchicken
    CHAOS GHOST!!!
    27 Oct '04 14:05
    Right, but that wouldn't make a very useful definition, because it doesn't recur or stop, and without a definition we'd have no way of calculating it anyway. So we say:

    e = exp(1) = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...
  12. Standard member PBE6
    Bananarama
    16 Nov '04 20:12
    Here's an easier way to get the answer:

    Let f(n) = (1+1/n)^n. Taking the natural logarithm of both sides we get:

    ln(f(n)) = ln[(1+1/n)^n] = n*ln(1+1/n)

    Now let x = 1/n:

    n*ln(1+1/n) = ln(1+x)/x

    Since lim(n-->inf) 1/n = 0, we can take the lim(x-->0) instead. Using L'Hopital's Rule, we get:

    lim(x-->0) ln(1+x)/x = lim(x-->0) ln(1+x)'/x' = lim(x-->0) (1/(1+x))/1

    lim(x-->0) (1/(1+x))/1 = (1/1)/1 = 1

    Therefore lim(x-->0) ln(f(n)) = 1. Taking the antilogarithm of both sides we get:

    lim(x-->0) f(n) = e

    Therefore:

    lim(n-->inf) f(n) = e







  13. Standard member The Plumber
    Leak-Proof
    16 Nov '04 23:48 / 1 edit
    Originally posted by !~TONY~!
    I figured it out today before the quiz. Good thing there was nothing of the sort on the test. Instead he unleashed this one. A ball is dropped from 6 feet above the ground, with each subsequent bounce being (3/4) that of the next. What is ...[text shortened]... traveled by the ball. I got it, but it took me about 15 minutes!
    42 feet

    (or 6 feet if you want to be a wise guy)