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Thread: Summation equality

  1. #1
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    Summation equality

    Summation equality-screenshot-2018-11-01-13.41.26.png

    I would like to understand how one gets to this equality. Thank you!
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  2. #2
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    Re: Summation equality

    Quote Originally Posted by mathfn View Post
    Click image for larger version. 

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    I would like to understand how one gets to this equality. Thank you!
    Please review the statement of the problem. There should have been some information given about $p$.
    It is a standard theorem that: if the function $f$ is integrable on $[a,b]$ and $c\in (a,b)$ then
    $\displaystyle \int_a^b f = \int_a^c f + \int_c^b f $.
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  3. #3
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    Re: Summation equality

    We just know that p>1. I am not sure if that helps.
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    Re: Summation equality

    Quote Originally Posted by mathfn View Post
    We just know that p>1. I am not sure if that helps.
    If $p>1$, then I do not believe the LHS will converge over the reals. Perhaps you are supposed to solve for $p$, and it is not greater than 1? In other words, you are looking for:

    $$\int_0^1 \dfrac{dx}{x^p} = 1$$

    This integral is fairly well-known. If $p\ge 1$, then you have the integral does not converge. If $p\neq 0$, then you have:

    $$\dfrac{1^{1-p}}{1-p}-\dfrac{0^{1-p}}{1-p}=1 \Longrightarrow p=0$$

    So, that means that $p=0$ is the only solution.
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    Re: Summation equality

    Quote Originally Posted by mathfn View Post
    We just know that p>1. I am not sure if that helps.
    Not only does it help, but it is necessary. Otherwise the statement would be false.
    Do you understand the theorem I quoted.

    We have $\displaystyle 0 < 1 < 2 < \cdots < n$ so by the theorem
    $\displaystyle \int_0^n {\frac{{dx}}{{{x^p}}}} = \int_0^1 {\frac{{dx}}{{{x^p}}}} + \int_1^2 {\frac{{dx}}{{{x^p}}}} + \cdots + \int_0^n {\frac{{dx}}{{{x^p}}}} = \sum\limits_{k = 1}^n {\int_{k - 1}^k {\frac{{dx}}{{{x^p}}}} } $

    Now $\displaystyle \int_0^1 {\frac{{dx}}{{{x^p}}}} = \mathop {\left. {\frac{{{x^{1 - p}}}}{{p - 1}}} \right|}\nolimits_0^1 =1$
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    Re: Summation equality

    Yes, I understand up to the last line. How do you know that the integral is equal to 1 without exactly knowing the value of p?
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  7. #7
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    Re: Summation equality

    Quote Originally Posted by mathfn View Post
    Yes, I understand up to the last line. How do you know that the integral is equal to 1 without exactly knowing the value of p?
    I agree, how is that known? I tried it for $p=2$ and according to Wolframalpha, the integral did not converge.

    Wolframalpha
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