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Thread: FUNdamental Theorem of calculus

  1. #1
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    FUNdamental Theorem of calculus

    Express the limit, $\displaystyle \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$ as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

    I have no idea what it's asking me to do haha
    And if your wondering I do know the the Fundamental Theorem of Calculus\
    Thanks : )
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    Quote Originally Posted by qzno View Post
    Express the limit, $\displaystyle \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$ as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

    I have no idea what it's asking me to do haha
    And if your wondering I do know the the Fundamental Theorem of Calculus\
    Thanks : )
    If you were to approximate the area under $\displaystyle y = e^x$ on $\displaystyle [0,1]$ using n rectangles of equal thickness you would use the following

    $\displaystyle \sum_{i=1}^n f(c_i) \Delta x,$

    where (if using the right endpoint of the subinterval)

    $\displaystyle c_i = \frac{i}{n},\;\;\; \Delta x = \frac{1}{n}$

    or

    $\displaystyle \sum_{i=1}^n e^{i/n} \, \frac{1}{n} \Delta x,$.

    In the limit as $\displaystyle n \to \infty$ the answer would becomes exact. So the answer would be

    $\displaystyle \int_0^1 e^x\, dx = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$.

    I'm guessing you can take it from here.
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    Quote Originally Posted by qzno View Post
    Express the limit, $\displaystyle \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$ as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

    I have no idea what it's asking me to do haha
    And if your wondering I do know the the Fundamental Theorem of Calculus\
    Thanks : )
    $\displaystyle \sum_{i=1}^n e^{\frac{i}{n}} \cdot \frac{1}{n} = (e^{\frac{1}{n}}+e^{\frac{2}{n}}+e^{\frac{3}{n}}+ ... + e^{1}) \cdot \frac{1}{n}$

    $\displaystyle \int_0^1 e^x dx$
    Last edited by skeeter; Jan 28th 2009 at 03:39 PM. Reason: ... too slow
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    why is it from 0 to 1?
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    look at the exponents for each term of the sum ...

    least term is $\displaystyle e^{\frac{1}{n}} \approx e^0$

    greatest term is $\displaystyle e^1$

    these are the y-values of each infinitesimal rectangle in the Riemann sum

    note also the factor $\displaystyle \frac{1}{n}$ ... a horizontal distance of 1 divided into n partitions.
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    Quote Originally Posted by qzno View Post
    why is it from 0 to 1?
    In general, the Riemann sum would be

    $\displaystyle \sum_{i=1}^n f(c_i)\, \Delta x$ where $\displaystyle \Delta x = \frac{b-a}{n}\;\;\; \text{and}\;\;\; c_i = a + \frac{b-a}{n}\;i $. Now compare with what you have

    $\displaystyle
    \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}
    $

    so $\displaystyle a + \frac{b-a}{n}\;i = \frac{i}{n}$ giving $\displaystyle a = 0,\;\;\; b = 1.$
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    thanks guys : )
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