I´d like to determine if this sequence converges and find out what its limit is.
$$c_n=\frac{1}{n^{s+1}}\sum_{k=1}^nk^s\qquad s>-1$$
Convergence of the sequence $\frac{1}{n^{s+1}}\sum_{k=1}^nk^s$, $s>-1$.
0
$\begingroup$
convergence
power-series
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0You mean as $n\to\infty$? – 2017-01-30
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0Is $*$ hadamard product.? – 2017-01-30
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0no, sry it is just multiplication – 2017-01-30
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1Note that $$c_n=\frac{1}{n}\sum \left(\frac{k}{n}\right)^s$$ is a Reimann sum for $\int_0^1 x^s\,dx$. – 2017-01-30
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0$$\int_0^1x^sdx$$ – 2017-01-30
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0but why? 0 and 1 come from the upper and lower limit of the fraction? – 2017-01-30
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0thank you very much, solved it – 2017-01-30
1 Answers
1
For $s > 0$ : $$k^s - \int_k^{k+1} x^s dx = \int_k^{k+1} (k^s- x^s) dx = -\int_k^{k+1} \int_k^x s t^{s-1}dt dx = \mathcal{O}(k^{s-1})$$ so that $$\sum_{k=1}^n k^{s} = \int_1^{n+1} x^s dx+\sum_{k=1}^n (k^{s}-\int_k^{k+1} x^s dx)$$ $$ = \frac{(n+1)^{s+1}-1}{s+1}+\sum_{k=1}^n \mathcal{O}(k^{s-1})= \frac{(n+1)^{s+1}-1}{s+1}+\mathcal{O}(n^s)$$
and for $s > -1$ it is the same with some $\mathcal{O}(1)$ terms