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lim-x-0-k-1-2021-k-x-2021-1-x-




Question Number 159405 by qaz last updated on 16/Nov/21
lim_(x→+0) (((Σ_(k=1) ^(2021) k^x )/(2021)))^(1/x) =?
$$\underset{\mathrm{x}\rightarrow+\mathrm{0}} {\mathrm{lim}}\left(\frac{\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{2021}} {\sum}}\mathrm{k}^{\mathrm{x}} }{\mathrm{2021}}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} =? \\ $$
Answered by mindispower last updated on 16/Nov/21
=lim_(x→0) e^((ln(((Σk^x )/(2021))))/x)   =e^((ln(Π_(k=1) ^(2021) k))/(2021)) =(2021!)^(1/(2021))
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{e}^{\frac{{ln}\left(\frac{\Sigma{k}^{{x}} }{\mathrm{2021}}\right)}{{x}}} \\ $$$$={e}^{\frac{{ln}\left(\underset{{k}=\mathrm{1}} {\overset{\mathrm{2021}} {\prod}}{k}\right)}{\mathrm{2021}}} =\left(\mathrm{2021}!\right)^{\frac{\mathrm{1}}{\mathrm{2021}}} \\ $$

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