Dirichlet's test explained

In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.^[1]

Statement

The test states that if

is a monotonic sequence of real numbers with $\backslash lim\_\; a\_n\; =\; 0$ and is a sequence of real numbers or complex numbers with bounded partial sums, then the series

$$\backslash sum\_^\; a\_n\; b\_n$$converges.

Proof

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

From summation by parts, we have that $S\_n\; =\; a\_\; B\_n\; +\; \backslash sum\_^\; B\_k\; (a\_k\; -\; a\_)$. Since the magnitudes of the partial sums

are bounded by some M and as , the first of these terms approaches zero: as .

Furthermore, for each k,

.

Since

is monotone, it is either decreasing or increasing:

• If

  (a_n)

  is decreasing,$$\backslash sum\_^n\; M|a\_k\; -\; a\_|\; =\; \backslash sum\_^n\; M(a\_k\; -\; a\_)\; =\; M\backslash sum\_^n\; (a\_k\; -\; a\_),$$which is a telescoping sum that equals

  M(a₁-a_n+1)

  and therefore approaches

  Ma₁

  as

  n\toinfty

  . Thus, $\backslash sum\_^\backslash infty\; M(a\_k\; -\; a\_)$ converges.
• If

  (a_n)

  is increasing,$$\backslash sum\_^n\; M|a\_k\; -\; a\_|\; =\; -\backslash sum\_^n\; M(a\_k\; -\; a\_)\; =\; -M\backslash sum\_^n\; (a\_k\; -\; a\_),$$which is again a telescoping sum that equals

  -M(a₁-a_n+1)

  and therefore approaches

  -Ma₁

  as

  n\toinfty

  . Thus, again, $\backslash sum\_^\backslash infty\; M(a\_k\; -\; a\_)$ converges.

So, the series $\backslash sum\_^\backslash infty\; B\_k(a\_k\; -\; a\_)$ converges by the direct comparison test to $\backslash sum\_^\backslash infty\; M(a\_k\; -\; a\_)$. Hence converges.

Applications

A particular case of Dirichlet's test is the more commonly used alternating series test for the case$$b\_n\; =\; (-1)^n\; \backslash Longrightarrow\backslash left|\backslash sum\_^N\; b\_n\backslash right|\; \backslash leq\; 1.$$

Another corollary is that $\backslash sum\_^\backslash infty\; a\_n\; \backslash sin\; n$ converges whenever

is a decreasing sequence that tends to zero. To see that is bounded, we can use the summation formula^[2] $$\backslash sum\_^N\backslash sin\; n=\backslash sum\_^N\backslash frac=\backslash frac=\backslash frac.$$

Improper integrals

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 non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral.

Notes

1. Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), pp. 253–255 . See also http://www.numdam.org/item/JMPA_1862_2_7__253_0/.
2. Web site: Where does the sum of $\sin(n)$ formula come from? .

References

• Book: Apostol, Tom M. . Tom M. Apostol . Calculus . John Wiley & Sons . 1967 . 0-471-00005-1 . 2nd . 1 . 1961.
• Hardy, G. H., A Course of Pure Mathematics, Ninth edition, Cambridge University Press, 1946. (pp. 379–380).
• Book: Rudin, Walter . Walter Rudin . Principles of mathematical analysis . McGraw-Hill . 1976 . 0-07-054235-X . 3rd . New York . 1502474 . 1953.
• Book: Spivak, Michael . Michael Spivak . Calculus . Publish or Perish . 2008 . 978-0-914098-91-1 . 4th . Houston, TX . 1967.
• Voxman, William L., Advanced Calculus: An Introduction to Modern Analysis, Marcel Dekker, Inc., New York, 1981. (§8.B.13–15) .

External links

• PlanetMath.org