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Math Help - sum of numbers

  1. #1
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    sum of numbers

    Hi My Friends , What's up ?

    The sum of squares :

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

    I have two questions :

    1- how do we derived the equation above ??

    2- what is the equation for the following sum :

    (1^0.25)+(2^0.25)+(3^0.25)+ ... +(n^0.25)

    Thank you very much
    Last edited by CaptainBlack; December 20th 2009 at 12:25 AM.
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  2. #2
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    Quote Originally Posted by metallica007 View Post
    Hi My Friends , What's up ?

    The sum of squares :

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

    I have two questions :

    1- how do we derived the equation above ??
    There are lots of ways of deriving the formula for the sum S_n of the squares of the first n positive integers (see the PlanetMath article). One way is to observe that the sum must be a cubic in n (this is obviously the case as the third differences of the sequence \{S_n, n=1, 2, ..\} are constant). Then we can use the first few terms of the sequence to find the coefficients of the cubic.

    2- what is the equation for the following sum :

    (1^0.25)+(2^0.25)+(3^0.25)+ ... +(n^0.25)
    I believe there is no elementary closed form for this (there is a formula in terms of generalised harmonic numbers or the polylogarithm function but these are not elementary).

    CB
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  3. #3
    Lord of certain Rings
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    There are various ways to solve that problem. An interesting one is to use geometric series:

    Let S_n = 1^2 + 2^2 + \cdot\cdot\cdot + n^2

    Let S = 1 + r + r^2 + \cdot\cdot\cdot + r^{n}
    \implies \frac{dS}{dr} = 1 + 2r +3r^2 + \cdot\cdot\cdot + nr^{n-1}
    \implies r\frac{dS}{dr} = r + 2r^2 +3r^3 + \cdot\cdot\cdot + nr^{n}
    \implies \frac{d}{dr}\left(r\frac{dS}{dr}\right) = 1 + 2^2r +3^2r^2 + \cdot\cdot\cdot + n^2r^{n-1}

    Therefore S_n = \frac{d}{dr}\left(r\frac{dS}{dr}\right)\bigg{|}_{r  =1}

    To evaluate the right hand side, use the fact that S = \frac{r^{n+1} - 1}{r - 1}
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  4. #4
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    Thank you very much my friends
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  5. #5
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    Code:
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