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] The test states that if
(
n
)
{\displaystyle (a_{n})}
is a monotonic sequence of real numbers with
lim
is a sequence of real numbers or complex numbers with bounded partial sums, then the series converges.
From summation by parts, we have that
Since the magnitudes of the partial sums
are bounded by some M and
, the first of these terms approaches zero:
is monotone, it is either decreasing or increasing: So, the series
converges by the direct comparison test to
converges.
[2][4] A particular case of Dirichlet's test is the more commonly used alternating series test for the case[2][5]
Another corollary is that
sin n
converges whenever
is a decreasing sequence that tends to zero.
sin n
is bounded, we can use the summation formula[6]
sin n =
sin 1 + sin
− sin (
2 − 2 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 non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral.