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Math Help - Given the integral e^x dx between a & b exists, evaluate it using the formula 1+r+r^2

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    Unhappy Given the integral e^x dx between a & b exists, evaluate it using the formula 1+r+r^2

    Given the integral e^x dx between a & b exists, evaluate it using the formula 1+r+r^2...r^n = 1 - r^(n+1)/(1-r), where r doesn't = 1.

    I tried, using (delta)x = (b-a)/n and xi = a + i(delta)x, but when I get to taking the limit I get stuck. I'm not even sure if that's what I'm suppose to do... but I'm not sure of any other way to go about it. Please help!
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by gummy_ratz View Post
    Given the integral e^x dx between a & b exists, evaluate it using the formula 1+r+r^2...r^n = 1 - r^(n+1)/(1-r), where r doesn't = 1.

    I tried, using (delta)x = (b-a)/n and xi = a + i(delta)x, but when I get to taking the limit I get stuck. I'm not even sure if that's what I'm suppose to do... but I'm not sure of any other way to go about it. Please help!
    What level is this? What definition of integral are you using?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    I think he (OP) meant using RIEMANN sums.

    Integration of e^x using riemann sums? - Yahoo! Answers
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    The course is Real Analysis. And we didn't evaluate any Riemann sums like that, but I'm guessing that must be what they want me to do. The question just says:
    "Given the integral between a & b e^x dx exists, evaluate it using the formula: 1 + r + r^2 + ... + r^n = (1-r^(n+1))/(1-r).

    I think that yahoo link might be something like what I need to do, but mine is just generally between a and b.

    And the definition of the integral I have is for every E>0 there's a d>0 : ||P|| < d -> |sigma - L| < E ... but I'm not really sure if I would apply that to this question?

    Do you guys know how I should go about solving this?
    Last edited by gummy_ratz; November 1st 2010 at 09:36 PM.
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    Divide the interval from a to b into n equal parts, each of length \Delta x= \frac{b- a}{n}. The value of the function f(x)= e^x at the left end of each sub-interval, is e^a, e^{a+ \Delta x}= e^a(e^{\Delta x}, e^{a+ 2\Delta x}= e^a(e^{2\Delta x})= e^a(e^{\Delta x})^2, etc.

    In other words, the integral is the limit of e^a\Delta x \left(1+ (e^{\Delta x})+ (e^{\Delta x})^2+ \cdot\cdot\cdot+ (e^{\Delta x})^n\right). Do you see what "r" must be?
    Last edited by mash; March 8th 2012 at 08:44 AM.
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    Okay great, thanks. Yeahh i get that. But where I run into trouble is when I try to take the limit.

    I keep having 1 - e^(delta x) on the bottom, which = 1-1 = 0 as n -> oo, and so I don't know how to get rid of that?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by gummy_ratz View Post
    Okay great, thanks. Yeahh i get that. But where I run into trouble is when I try to take the limit.

    I keep having 1 - e^(delta x) on the bottom, which = 1-1 = 0 as n -> oo, and so I don't know how to get rid of that?
    Try multiplying top and bottom by $latex e^{\Delta x}$
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    Quote Originally Posted by Drexel28 View Post
    Try multiplying top and bottom by $latex e^{\Delta x}$
    But I think e^(delta x) = 1 as x-> 00, since

    delta x = (b-a)/n = 0 as x-> 00 and e^0 = 1

    so that would just be multiplying the top and bottom by 1?

    I have:

    lim x->00 = e^(a+deltax)*(1- e^(deltax*n+deltax))/(1-e^delax)

    but then even when I factor e^deltax out of the top and bottom, I'm still left with
    ((1/e^deltax) - 1) on the bottom.
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