1. Donation!~TONY~!
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    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 membertelerion
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    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. SubscriberAThousandYoung
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    26 Oct '04 08:141 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. SubscriberAThousandYoung
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    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. SubscriberAThousandYoung
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    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 memberroyalchicken
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    26 Oct '04 11:241 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~!
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    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 membertelerion
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    26 Oct '04 17:081 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~!
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    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....
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    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 memberroyalchicken
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    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 memberPBE6
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    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 memberThe Plumber
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    16 Nov '04 23:481 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)😉
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