Integral test for convergence

Proof

The proof uses the comparison test, comparing the term ${\displaystyle f(n)}$ with the integral of ${\displaystyle f}$ over the intervals ${\displaystyle [n-1,n)}$ and ${\displaystyle [n,n+1)}$ respectively.

The monotonic function ${\displaystyle f}$ is continuous almost everywhere. To show this, let

${\displaystyle D=\{x\in [N,\mathrm{\infty })\mid f\text{ is discontinuous at }x\}}$

For every ${\displaystyle x\in D}$, there exists by the density of ${\displaystyle \mathbb{Q}}$, a ${\displaystyle c(x)\in \mathbb{Q}}$ so that ${\displaystyle c(x)\in \left[\underset{y\downarrow x}{lim}f(y),\underset{y\uparrow x}{lim}f(y)\right]}$.

Note that this set contains an open non-empty interval precisely if ${\displaystyle f}$ is discontinuous at ${\displaystyle x}$. We can uniquely identify ${\displaystyle c(x)}$ as the rational number that has the least index in an enumeration ${\displaystyle \mathbb{N}\to \mathbb{Q}}$ and satisfies the above property. Since ${\displaystyle f}$ is monotone, this defines an injective mapping ${\displaystyle c:D\to \mathbb{Q},x\mapsto c(x)}$ and thus ${\displaystyle D}$ is countable. It follows that ${\displaystyle f}$ is continuous almost everywhere. This is sufficient for Riemann integrability.

Since f is a monotone decreasing function, we know that

${\displaystyle f(x)\le f(n)\text{for all }x\in [n,\mathrm{\infty })}$

and

${\displaystyle f(n)\le f(x)\text{for all }x\in [N,n].}$

Hence, for every integer n ≥ N,

and, for every integer n ≥ N + 1,

By summation over all n from N to some larger integer M, we get from (2)

${\displaystyle {\int }_N^{M+1}f(x)dx=\sum _{n=N}^M\underset{\le f(n)}{\underbrace{{\int }_n^{n+1}f(x)dx}}\le \sum _{n=N}^{M}f(n)}$

and from (3)

${\displaystyle \begin{array}{cc}\sum _{n=N}^{M}f(n)& =f(N)+\sum _{n=N+1}^{M}f(n)\\ & \le f(N)+\sum _{n=N+1}^M\underset{\ge f(n)}{\underbrace{{\int }_{n-1}^{n}f(x)dx}}\\ & =f(N)+{\int }_N^{M}f(x)dx.\end{array}}$

Combining these two estimates yields

${\displaystyle {\int }_N^{M+1}f(x)dx\le \sum _{n=N}^{M}f(n)\le f(N)+{\int }_N^{M}f(x)dx.}$

Letting M tend to infinity, the bounds in (1) and the result follow.

Applications

The harmonic series

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}}$

diverges because, using the natural logarithm, its antiderivative, and the fundamental theorem of calculus, we get

${\displaystyle {\int }_1^M\frac{1}{n}dn=\ln n{|}_1^M=\ln M\to \mathrm{\infty }\text{for }M\to \mathrm{\infty }.}$

On the other hand, the series

${\displaystyle \zeta (1+\epsilon )=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1+\epsilon }}}$

(cf. Riemann zeta function) converges for every ε > 0, because by the power rule

${\displaystyle {\int }_1^M\frac{1}{n^{1+\epsilon }}dn={-\frac{1}{\epsilon n^{\epsilon }}|}_1^M=\frac{1}{\epsilon }\left(1-\frac{1}{M^{\epsilon }}\right)\le \frac{1}{\epsilon }<\mathrm{\infty }\text{for all }M\ge 1.}$

From (1) we get the upper estimate

${\displaystyle \zeta (1+\epsilon )=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1+\epsilon }}\le \frac{1+\epsilon }{\epsilon },}$

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 f(n) decreases to 0 faster than 1/n but slower than 1/n^1+ε in the sense that

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{f(n)}{1/n}=0\text{and}\underset{n\to \mathrm{\infty }}{lim}\frac{f(n)}{1/n^{1+\epsilon }}=\mathrm{\infty }}$

for every ε > 0, and whether the corresponding series of the f(n) still diverges. Once such a sequence is found, a similar question can be asked with f(n) taking the role of either 1/n (if f(n) diverges) or 1/n^1+ε (if f(n) converges), 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 k, the series

still diverges (cf. proof that the sum of the reciprocals of the primes diverges for k = 1) but

converges for every ε > 0. Here ln_k denotes the k-fold composition of the natural logarithm defined recursively by

${\displaystyle {\ln }_k(x)=\{\begin{array}{ll}\ln (x)& \text{for }k=1,\\ \ln ({\ln }_{k-1}(x))& \text{for }k\ge 2.\end{array}}$

Furthermore, let N_k denote the smallest natural number such that the k-fold composition is well-defined and ln_k(N_k) ≥ 1, i.e.

${\displaystyle N_k\ge \underset{ke\text{s}}{\underbrace{e^{e^{{\cdot }^{{\cdot }^e}}}}}=e\uparrow <!--\; -->\uparrow k}$

using tetration or Knuth's up-arrow notation.

To see the divergence of the series (4) using the integral test, note that by repeated application of the chain rule

${\displaystyle \frac{d}{dx}{\ln }_{k+1}(x)=\frac{d}{dx}\ln ({\ln }_k(x))=\frac{1}{{\ln }_k(x)}\frac{d}{dx}{\ln }_k(x)=\cdots =\frac{1}{x\ln (x)\cdots {\ln }_k(x)},}$

hence

${\displaystyle {\int }_{N_k}^{\mathrm{\infty }}\frac{dx}{x\ln (x)\cdots {\ln }_k(x)}={\ln }_{k+1}(x){|}_{N_k}^{\mathrm{\infty }}=\mathrm{\infty }.}$

To see the convergence of the series (5), note that by the power rule, the chain rule, and the above result,

${\displaystyle -\frac{d}{dx}\frac{1}{\epsilon ({\ln }_k(x){)}^{\epsilon }}=\frac{1}{({\ln }_k(x){)}^{1+\epsilon }}\frac{d}{dx}{\ln }_k(x)=\cdots =\frac{1}{x\ln (x)\cdots {\ln }_{k-1}(x)({\ln }_k(x){)}^{1+\epsilon }},}$

hence

${\displaystyle {\int }_{N_k}^{\mathrm{\infty }}\frac{dx}{x\ln (x)\cdots {\ln }_{k-1}(x)({\ln }_k(x){)}^{1+\epsilon }}=-\frac{1}{\epsilon ({\ln }_k(x){)}^{\epsilon }}{|}_{N_k}^{\mathrm{\infty }}<\mathrm{\infty }}$

and (1) gives bounds for the infinite series in (5).