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I know that

$\displaystyle \sum_{n=1}^\infty x^n = \frac{1}{1-x}$ (Geometric series)

and that the harmonic series is divergent:

$\displaystyle \sum_{n=1}^\infty \frac{1}{n} \rightarrow \infty$

And I see quite often series of this form:

$s_\varepsilon =\displaystyle \sum_{n=1}^\infty \frac{1}{n^{1+\varepsilon}}$ with $\varepsilon > 0$

I know that $s_{\varepsilon > 0}$ converges due to the root test. But what is the value of those series?

So here is my question:

Let $\varepsilon > 0$. What is the value of $s_\varepsilon := \displaystyle \sum_{n=1}^\infty \frac{1}{n^{1+\varepsilon}}$?

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    Side note: The root test doesn't help here ($\limsup_{n\to\infty}\sqrt[n]{|a_n|} = 1$). In fact, this is mentioned in the Wikipedia article you linked to.2012-09-21
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    $s_\epsilon \rightarrow \frac1\epsilon$2012-09-21
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    @karakfa: What do you mean?2012-09-21
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    @karakfa: Does your answer mean anything else than "$s_\varepsilon$ diverges for $\varepsilon \rightarrow 0$"?2012-09-21
  • 0
    Well, but many systems will have numerical algorithms for the zeta function ...2012-09-21
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    The value of that sum is the Riemann zeta function evaluated at $1+\varepsilon$.2012-09-21

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