lim_(n→∞) Σ_(k=1) ^n ((1/k))^k


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Question Number 167332 by Amazigh last updated on 13/Mar/22

$\lim_{n \to \infty} \underset{k = 1}{\sum^{n}} {(\frac{1}{k})}^{k}$
	Answered by LEKOUMA last updated on 14/Mar/22

	$S_{n} = \lim_{n \to 0} \underset{k = 1}{\sum^{n}} {(\frac{1}{k})}^{k}$ $O n p r e n d, a = 0 e t b = 1$ $C h o i s i s o n f (x) = x^{k}$ $f e s t c o n t i n u e s u r [0, 1]$ $S_{n} \to \int_{0}^{1} f (x) d x = \int_{0}^{1} x^{k} d x$ $\int_{0}^{1} x^{k} d x = {[\frac{x^{k + 1}}{k + 1}]}_{0}^{1} = \frac{1}{k + 1}$ $\lim_{n \to \infty} \underset{k = 1}{\sum^{n}} {(\frac{1}{k})}^{k} = \frac{1}{k + 1}$
	Commented by Amazigh last updated on 17/Mar/22

	$P o u r q o i S_{n} = \lim_{x \to 0} \underset{k = 1}{\sum^{n}} {(\frac{1}{k})}^{k}$ $e t l a l i m i t e n e d o i t p a s d e p e n d r e d e k (\frac{1}{k + 1}), c^{'} e s t u n e s e r i e e n t i e r e q u i c o n v e r g e .$
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