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Math Help - Power mean

  1. #1
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    Power mean

    From the power mean inequality  P_r = \left(\frac{a_{1}^{r} + \ldots + a_{n}^{r} }{n}\right)^{1/r} why does  \lim_{r \to 0} P_{r} = (a_{1}a_{2} \cdots a_{n})^{1/n} ?
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  2. #2
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    Quote Originally Posted by particlejohn View Post
    From the power mean inequality  P_r = \left(\frac{a_{1}^{r} + \ldots + a_{n}^{r} }{n}\right)^{1/r} why does  \lim_{r \to 0} P_{r} = (a_{1}a_{2} \cdots a_{n})^{1/n} ?
    There's a proof in this link: Generalized mean - Wikipedia, the free encyclopedia
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  3. #3
    Super Member PaulRS's Avatar
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    First: <br />
\mathop {\lim }\limits_{r \to 0} \log \left( {P_r } \right) = \mathop {\lim }\limits_{r \to 0} \tfrac{1}<br />
{r} \cdot \log \left( {\tfrac{{\sum\nolimits_{k = 1}^n {a_k ^r } }}<br />
{n}} \right)<br />

    Now we have: <br />
\log \left( {\tfrac{{\sum\nolimits_{k = 1}^n {a_k ^r } }}<br />
{n}} \right)\mathop  \sim \limits_{r \to 0} \tfrac{{\sum\nolimits_{k = 1}^n {a_k ^r } }}<br />
{n} - 1 = \tfrac{{\sum\nolimits_{k = 1}^n {\left( {a_k ^r  - 1} \right)} }}<br />
{n}<br />
(since what's inside the logarithm tends to 1)

    So: <br />
\mathop {\lim }\limits_{r \to 0} \log \left( {P_r } \right) = \mathop {\lim }\limits_{r \to 0} \tfrac{1}<br />
{r} \cdot \tfrac{{\sum\nolimits_{k = 1}^n {\left( {a_k ^r  - 1} \right)} }}<br />
{n} = \tfrac{1}<br />
{n} \cdot \sum\nolimits_{k = 1}^n {\mathop {\lim }\limits_{r \to 0} \tfrac{1}<br />
{r} \cdot \left( {a_k ^r  - 1} \right)} <br />
(by the linearity of the limit- all of those are defined-)

    <br />
\mathop {\lim }\limits_{r \to 0} \tfrac{1}<br />
{r} \cdot \left( {a_k ^r  - 1} \right) = \mathop {\lim }\limits_{r \to 0} \tfrac{1}<br />
{r} \cdot \left( {e^{r \cdot \log \left( {a_k } \right)}  - 1} \right) = \log \left( {a_k } \right)<br />

    So: <br />
\mathop {\lim }\limits_{r \to 0} \log \left( {P_r } \right) = \tfrac{1}<br />
{n} \cdot \sum\nolimits_{k = 1}^n {\log \left( {a_k } \right)} <br />
and the rest follows easily by the properties of the logarithm and the continuity of the exponential function.
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