Editing Integral test for convergence (section)

= \left. -\frac 1{\varepsilon n^\varepsilon} \right|_1^M=
\frac 1 \varepsilon \left(1-\frac 1 {M^\varepsilon}\right)
\le \frac 1 \varepsilon  < \infty
\quad\text{for all }M\ge1.
</math>
From ({{EquationNote|1}}) we get the upper estimate
:<math>
\zeta(1+\varepsilon)=\sum_{n=1}^\infty \frac 1 {n^{1+\varepsilon}} \le \frac{1 + \varepsilon}\varepsilon,
</math>
which can be compared with some of the [[particular values of Riemann zeta function]].

==Borderline between divergence and convergence==
The above examples involving the harmonic series raise the question of whether there are monotone sequences such that {{math|''f''(''n'')}} decreases to 0 faster than {{math|1/''n''}} but slower than {{math|1/''n''<sup>1+''ε''</sup>}} in the sense that
:<math>
\lim_{n\to\infty}\frac{f(n)}{1/n}=0
\quad\text{and}\quad
\lim_{n\to\infty}\frac{f(n)}{1/n^{1+\varepsilon}}=\infty
</math>
for every {{math|''ε'' > 0}}, and whether the corresponding series of the {{math|''f''(''n'')}} still diverges. Once such a sequence is found, a similar question can be asked with {{math|''f''(''n'')}} taking the role of {{math|1/''n''}}, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series.

Using the integral test for convergence, one can show (see below) that, for every [[natural number]] {{math|''k''}}, the series
{{NumBlk|:|<math>
\sum_{n=N_k}^\infty\frac1{n\ln(n)\ln_2(n)\cdots \ln_{k-1}(n)\ln_k(n)}
</math>|{{EquationRef|4}}}}
still diverges (cf. [[proof that the sum of the reciprocals of the primes diverges]] for {{math|''k'' {{=}} 1}}) but
{{NumBlk|:|<math>
\sum_{n=N_k}^\infty\frac1{n\ln(n)\ln_2(n)\cdots\ln_{k-1}(n)(\ln_k(n))^{1+\varepsilon}}
</math>|{{EquationRef|5}}}}
converges for every {{math|''ε'' > 0}}. Here {{math|ln<sub>''k''</sub>}} denotes the {{math|''k''}}-fold [[function composition|composition]] of the natural logarithm defined [[recursion|recursively]] by
:<math>
\ln_k(x)=
\begin{cases}
\ln(x)&\text{for }k=1,\\
\ln(\ln_{k-1}(x))&\text{for }k\ge2.
\end{cases}
</math>
Furthermore, {{math|''N''<sub>''k''</sub>}} denotes the smallest natural number such that the {{math|''k''}}-fold composition is well-defined and {{math|ln<sub>''k''</sub>(''N''<sub>''k''</sub>) ≥ 1}}, i.e.
:<math>
N_k\ge \underbrace{e^{e^{\cdot^{\cdot^{e}}}}}_{k\ e'\text{s}}=e \uparrow\uparrow k
</math>
using [[tetration]] or [[Knuth's up-arrow notation]].

To see the divergence of the series ({{EquationNote|4}}) using the integral test, note that by repeated application of the [[chain rule]]
:<math>
\frac{d}{dx}\ln_{k+1}(x)
=\frac{d}{dx}\ln(\ln_k(x))
=\frac1{\ln_k(x)}\frac{d}{dx}\ln_k(x)
=\cdots
=\frac1{x\ln(x)\cdots\ln_k(x)},
</math>
hence
:<math>
\int_{N_k}^\infty\frac{dx}{x\ln(x)\cdots\ln_k(x)}
=\ln_{k+1}(x)\bigr|_{N_k}^\infty=\infty.
</math>
To see the convergence of the series ({{EquationNote|5}}), note that by the [[power rule]], the chain rule and the above result
:<math>
-\frac{d}{dx}\frac1{\varepsilon(\ln_k(x))^\varepsilon}
=\frac1{(\ln_k(x))^{1+\varepsilon}}\frac{d}{dx}\ln_k(x)
=\cdots
=\frac{1}{x\ln(x)\cdots\ln_{k-1}(x)(\ln_k(x))^{1+\varepsilon}},
</math>
hence
:<math>
\int_{N_k}^\infty\frac{dx}{x\ln(x)\cdots\ln_{k-1}(x)(\ln_k(x))^{1+\varepsilon}}
=-\frac1{\varepsilon(\ln_k(x))^\varepsilon}\biggr|_{N_k}^\infty<\infty
</math>
and ({{EquationNote|1}}) gives bounds for the infinite series in ({{EquationNote|5}}).

==See also==
*[[Convergence tests]]
*[[Convergence (mathematics)]]
*[[Direct comparison test]]
*[[Dominated convergence theorem]]
*[[Euler-Maclaurin formula]]
*[[Limit comparison test]]
*[[Monotone convergence theorem]]

==References==
* [[Konrad Knopp|Knopp, Konrad]], "Infinite Sequences and Series", [[Dover Publications]], Inc.,  New York, 1956. (§ 3.3) {{ISBN|0-486-60153-6}}
* [[Whittaker and Watson|Whittaker, E. T., and Watson, G. N., ''A Course in Modern Analysis'']], fourth edition, Cambridge University Press, 1963. (§ 4.43) {{ISBN|0-521-58807-3}}
* Ferreira, Jaime Campos, Ed Calouste Gulbenkian, 1987,  {{ISBN|972-31-0179-3}}
<references/>

{{Calculus topics}}

[[Category:Augustin-Louis Cauchy]]
[[Category:Integral calculus]]
[[Category:Convergence tests]]
[[Category:Articles containing proofs]]