Όριο αθροίσματος (3)

#1

Έστω $\displaystyle{p\in\mathbb N^{*}}$ . Ας βρεθεί, αν υπάρχει, το όριο $\displaystyle{\lim_{n\to+\infty}\left(\frac{1}{p}\sum_{k=1}^{p}k^{\frac{p}{n}}\right)^{n}}$ .

#2

$\displaystyle{\mathop {\lim }\limits_{x \to 0 + } \frac{{\ln \left( {{1^{p \cdot x}} + {2^{p \cdot x}} + {\text{ }}..{\text{ }} + {p^{p \cdot x}}} \right) - \ln \left( p \right)}}{x} = \mathop = \limits^{\left( {\dfrac{0}{0}} \right)} = \mathop {\lim }\limits_{x \to 0 + } \frac{{p \cdot \ln \left( 2 \right) \cdot {2^{p \cdot x}} + {\text{ }}..{\text{ }} + p \cdot \ln \left( p \right) \cdot {2^{p \cdot x}}}}{{{1^x} + {2^{2 \cdot x}} + .. + {p^{p \cdot x}}}} = }$

$\displaystyle{ = \frac{{p \cdot \ln \left( {p!} \right)}}{p} = \ln \left( {p!} \right) \Rightarrow \mathop {\lim }\limits_{x \to 0 + } {\left( {\frac{{{1^{p \cdot x}} + {2^{p \cdot x}} + {\text{ }}..{\text{ }} + {p^{p \cdot x}}}}{p}} \right)^{\frac{1}{x}}} = \mathop {\lim }\limits_{x \to 0 + } {e^{\dfrac{{\ln \left( {{1^{p \cdot x}} + {2^{p \cdot x}} + {\text{ }}..{\text{ }} + {p^{p \cdot x}}} \right) - \ln \left( p \right)}}{x}}} = {e^{\ln \left( {p!} \right)}} = p!}$ .

Τότε $\displaystyle{\mathop {\lim }\limits_{x \to + \infty } {\left( {\frac{{{1^{\dfrac{p}{x}}} + {2^{\dfrac{p}{x}}} + {\text{ }}..{\text{ }} + {p^{\dfrac{p}{x}}}}}{p}} \right)^\big{x}} = \mathop {\lim }\limits_{x \to 0 + } {\left( {\frac{{{1^\big{{p \cdot x}}} + {2^\big{{p \cdot x}}} + {\text{ }}..{\text{ }} + {p^\big{{p \cdot x}}}}}{p}} \right)^{\dfrac{1}{x}}} = p!}$

και τελικά $\displaystyle{\mathop {\lim }\limits_{x \to + \infty } {\left( {\frac{1}{p} \cdot \sum\limits_{k = 1}^p {{k^{\dfrac{p}{n}}}} } \right)^\big{n}} = p!}$

Όριο αθροίσματος (3)

Όριο αθροίσματος (3)

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