Dirichlet's test#Improper integrals

The test states that if $(a\_n)$ is a monotonic sequence of real numbers with $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; a\_n\; =\; 0$ and $(b\_n)$ is a sequence of real numbers or complex numbers with bounded partial sums, then the series

:$$\backslash sum\_\{n=1\}^\{\backslash infty\}\; a\_n\; b\_n$$

converges.{{harvnb|Apostol|1967|pp=407–409}}{{harvnb|Spivak|2008|p=495}}{{Harvnb|Rudin|1976|p=70}}

Let $S\_n\; =\; \backslash sum\_\{k=1\}^n\; a\_k\; b\_k$ and $B\_n\; =\; \backslash sum\_\{k=1\}^n\; b\_k$.

From summation by parts, we have that $S\_n\; =\; a\_\{n\}\; B\_n\; +\; \backslash sum\_\{k=1\}^\{n-1\}\; B\_k\; (a\_k\; -\; a\_\{k+1\})$. Since the magnitudes of the partial sums $B\_n$ are bounded by some M and $a\_n\; \backslash to\; 0$ as $n\backslash to\backslash infty$, the first of these terms approaches zero: $|a\_\{n\}\; B\_n|\; \backslash leq\; |a\_\{n\}\; M|\; \backslash to\; 0$ as $n\backslash to\backslash infty$.

Furthermore, for each k, $|B\_k\; (a\_k\; -\; a\_\{k+1\})|\; \backslash leq\; M|a\_k\; -\; a\_\{k+1\}|$.

Since $(a\_n)$ is monotone, it is either decreasing or increasing:

• If $(a\_n)$ is decreasing,

  $$\backslash sum\_\{k=1\}^n\; M|a\_k\; -\; a\_\{k+1\}|\; =\; \backslash sum\_\{k=1\}^n\; M(a\_k\; -\; a\_\{k+1\})\; =\; M\backslash sum\_\{k=1\}^n\; (a\_k\; -\; a\_\{k+1\}),$$

  which is a telescoping sum that equals $M(a\_1\; -\; a\_\{n+1\})$ and therefore approaches $Ma\_1$ as $n\; \backslash to\; \backslash infty$. Thus, $\backslash sum\_\{k=1\}^\backslash infty\; M(a\_k\; -\; a\_\{k+1\})$ converges.

• If $(a\_n)$ is increasing,

  $$\backslash sum\_\{k=1\}^n\; M|a\_k\; -\; a\_\{k+1\}|\; =\; -\backslash sum\_\{k=1\}^n\; M(a\_k\; -\; a\_\{k+1\})\; =\; -M\backslash sum\_\{k=1\}^n\; (a\_k\; -\; a\_\{k+1\}),$$

  which is again a telescoping sum that equals $-M(a\_1\; -\; a\_\{n+1\})$ and therefore approaches $-Ma\_1$ as $n\backslash to\backslash infty$. Thus, again, $\backslash sum\_\{k=1\}^\backslash infty\; M(a\_k\; -\; a\_\{k+1\})$ converges.

So, the series $\backslash sum\_\{k=1\}^\backslash infty\; B\_k(a\_k\; -\; a\_\{k+1\})$ converges by the direct comparison test to $\backslash sum\_\{k=1\}^\backslash infty\; M(a\_k\; -\; a\_\{k+1\})$. Hence $S\_n$ converges.

A particular case of Dirichlet's test is the more commonly used alternating series test for the case{{Harvnb|Rudin|1976|p=71}}

$$b\_n\; =\; (-1)^n\; \backslash Longrightarrow\backslash left|\backslash sum\_\{n=1\}^N\; b\_n\backslash right|\; \backslash leq\; 1.$$

Another corollary is that $\backslash sum\_\{n=1\}^\backslash infty\; a\_n\; \backslash sin\; n$ converges whenever $(a\_n)$ is a decreasing sequence that tends to zero. To see that

$\backslash sum\_\{n=1\}^N\; \backslash sin\; n$

is bounded, we can use the summation formula{{cite web | url=https://math.stackexchange.com/q/1119043 | title=Where does the sum of $\sin(n)$ formula come from? }}

$$\backslash sum\_\{n=1\}^N\backslash sin\; n=\backslash sum\_\{n=1\}^N\backslash frac\{e^\{in\}-e^\{-in\}\}\{2i\}=\backslash frac\{\backslash sum\_\{n=1\}^N\; (e^\{i\})^n-\backslash sum\_\{n=1\}^N\; (e^\{-i\})^n\}\{2i\}=\backslash frac\{\backslash sin\; 1\; +\backslash sin\; N-\backslash sin\; (N+1)\}\{2-\; 2\backslash cos\; 1\}.$$

An analogous statement for convergence of improper integrals is proven using integration by parts. If the integral of a function f is uniformly bounded over all intervals, and g is a bounded non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral.

• {{Cite book |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |title=Calculus |publisher=John Wiley & Sons |year=1967 |isbn=0-471-00005-1 |edition=2nd |volume=1 |orig-year=1961}}
• Hardy, G. H., A Course of Pure Mathematics, Ninth edition, Cambridge University Press, 1946. (pp. 379–380).
• {{Cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Principles of mathematical analysis |publisher=McGraw-Hill |year=1976 |isbn=0-07-054235-X |edition=3rd |location=New York |oclc=1502474 |orig-date=1953}}
• {{Cite book |last=Spivak |first=Michael |author-link=Michael Spivak |title=Calculus |publisher=Publish or Perish |year=2008 |isbn=978-0-914098-91-1 |edition=4th |location=Houston, TX |orig-year=1967}}
• Voxman, William L., Advanced Calculus: An Introduction to Modern Analysis, Marcel Dekker, Inc., New York, 1981. (§8.B.13–15) {{ISBN|0-8247-6949-X}}.

• [https://planetmath.org/dirichletsconvergencetest PlanetMath.org]

{{Calculus topics}}

Category:Convergence tests