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Dirichlet’s Test

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Series Convergence Tests >

Dirichlet’s test is one way to determine if an infinite series converges to a finite value. The test, named after 19th-century German mathematician Peter Gustav Lejeune Dirichlet, is a generalization of the alternating series test.

Formally, Dirichlet’s test states that the infinite series
a1b1 + a2b2 + … + anbn converges if the following two statements are true:


  1. The sequence of partial sums
    sn = a1 + a2 + … an
    is a bounded sequence. In other words, there is a positive number K so that
    Sn < K for all n.
  2. b1 + b2 + … bn is a monotonic decreasing sequence (i.e. a steadily decreasing sequence) converging to zero (i.e. bn < bn-1 and limn→∞) bn = 0).

When to Use Dirichlet’s Test

Dirichlet’s test is one of the lesser known tests. In general, the common rules for convergence of series—the ones you learn in elementary calculus—suffice for testing the vast majority of series. But there are some specific cases where the “usual” tests just don’t work.

For example, you can use the ratio test or root test to show that the following power series diverges (for |z|> 1) or converges absolutely for |z| < 1.
dirichlets test example

However, neither of those tests tell you what happens when z = 1. For that, you can use Dirichlet’s test to show that the series converges (Evans, 2009).

Example of Dirichlet’s Test

Use Dirchlet’s test to show that the following series converges:

Step 1: Rewrite the series into the form a1b1 + a2b2 + … + anbn:

Step 2: Show that the sequence of partial sums an is bounded. One way to tackle this to to evaluate the first few sums and see if there is a trend:


  • a2 = cos(2π) = 1
  • a3 = cos(2π) + cos(3π) = 1 – 1 = 0
  • a4 = cos(π) + cos(2π) + cos(3π) = 1 – 1 + 1 = 0

It appears the sequence of partial sums is bounded (≤1).

Step 3:Evaluate bn to see if it decreasing. One way to do this is to graph the function (I used Desmos.com):


Clearly, the function (and therefore the sequence) is decreasing and the limit as n→∞ is 0. Therefore, this series converges.

Proof of Dirchlet’s Test

Watch the following video for a proof of convergence using Dirchlet’s test:

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References

Clapham, C. & Nicholson, J. (2014). The Concise Oxford Dictionary of Mathematics. OUP Oxford.
Evans, P. (2009). Math 140A Test 2. Retrieved September 18, 2020 from: http://math.ucsd.edu/~lni/math140/math140a_Midterm_Sample2.pdf
Mathonline. Dirichlet’s Test for Convergence of Series of Real Numbers Examples 1. Retrieved September 18, 2020 from: http://mathonline.wikidot.com/dirichlet-s-test-for-convergence-examples-1
Nelson, D. (2008). The Penguin Dictionary of Mathematics. Penguin Books Limited.

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Stephanie Glen. "Dirichlet’s Test" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/dirichlets-test/
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