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 $s_\varepsilon := \displaystyle \sum_{n=1}^\infty \frac{1}{n^{1+\varepsilon}}$?