Series Convergence Tests

Series Convergence Tests Contents (Click to skip to that section):

  1. Series Convergence Tests in Alphabetical Order
  2. What Does “Converge” Mean?
  3. What does “Diverge” Mean?

Series Convergence Tests in Alphabetical Order

Often, you’ll want to know whether a series converges (i.e. reaches a certain number) or diverges (does not converge). Figuring this out from scratch can be an extremely difficult task —something that’s beyond the scope of even a calculus II course. Thankfully, mathematicians before you have calculated Series Convergence Tests: the convergence or divergence of many common series. This enables you to figure out whether a particular series may or may not converge.

Abel’s Test

Abel’s test is a convergence test for infinite series; It tells us whether some infinite series converges in certain situations.

More info: Abel’s test.

Absolute Convergence

If the absolute value of the series
absolute convergence

converges, then the series
absolute convergence test

Alternating Series Convergence Tests

If for all n, an is positive, non-increasing (i.e. 0 < = an) and approaches 0, then the alternating series test tells us that the following alternating series converges:
Alternating Series Convergence Tests

If the series converges, then the remainder R,sub>N = S – SN is bounded by |R N|< = aN + 1. S is the exact sum of the infinite series and SN is the sum of the first N terms of the series.

Deleting the first N Terms

The following series either both converge or both diverge if N is a positive integer.
Deleting the first N Terms

Series Convergence Tests: Dirichlet’s Test

Dirichlet’s test is a generalization of the alternating series test.

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

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 Dirichlet’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

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:

Proof that sum(sin(n)/n) Converges using Dirichlet's Test

Direct Comparison Test

In the direct comparison test, the following two rules apply if 0 < = an &lt ;= bn for all n greater than some positive integer N.
Direct Comparison Test

Geometric Series Convergence Tests

Geometric Series Convergence Tests

With the geometric series, if r is between -1 and 1 then the series converges to 1(1 – r).

Integral Series Convergence Tests

The following series either both converge or both diverge if, for all n> = 1, f(n) = an and f is positive, continuous and decreasing. If the series does converge, then the remainder RN is bounded by
Integral Series Convergence Tests

See: Integral Series / Remainder Estimate.

Limit Comparison Test

The limit comparison test states that the following series either both converge or both diverge if lim(N → ∞) (anbn where an,bn>0 and L is positive and finite.

nth-Term Test for Divergence

The following series diverges if the sequence {an} doesn’t converge to 0:

For several examples, see: Nth Term Test for Divergence.

P series

If p > 1, then the p-series converges.
If 0 < p < 1 then the series diverges.

Ratio Test

The following rules apply if for all n, n≠0. L = lim (n→ ∞)|an + 1an|.
If L<1, then the series


If L>1, then the series

If L = 1, then the ratio test is inconclusive.

More info: Ratio Test

Root Test

The root test tells us whether or not a series converges absolutely.
As its name suggests, it involves taking a root, and so it is most useful for exponential series and in situations where taking a root is quick and simple.

Determining Absolute Convergence Using the Root Test

To use the root test on a series:
use the root test on a series

First, calculate the limit:
root test

  • If that limit is less than one, the series converges absolutely.
  • If it is greater than one, it diverges.
  • And if the limit is just one, we don’t know. It might converge conditionally, it might diverge.

There are other tests which can be used to determine convergence when the root test gives us an indeterminate answer.

Examples of the Root Test

To see how this works, suppose you are given the series:

We need to take the limit, as n goes to infinity, of the nth root of that. Taking the nth root divides any exponent by 1/n. In the above formula, there are two exponents (n in the numerator and 1 + 2n in the denominator); each of these needs to be divided by 1/n.

Using that information, you should get:
root test

Since infinity is greater than 1, this series is divergent.

The following series is a little more interesting:

Here, again, we divide the exponents by 1/n.

root test

Since 12/10 is also greater than one, this series also diverges.


Belk, Jim. Math 142 Course Notes. Root Test. Bard College Math Department. Published Online Fall 2009. Retrieved from on August 26, 2019.
Oregon State Calculus Quest. Retrieved from on August 26, 2019

Taylor Series Convergence

The Taylor series converges if f has derivatives of all orders on an interval “I” centered at c, if lim(n→ infin;)RN = 0 for all x in l:

The Taylor series remainder of RN = S – SN is equal to (1/(n + 1)!)f(n + 1)(z)(x – c)n + 1 where z is a constant between x and c.

What Does “Converge” Mean?

Converge means to settle on a certain number. For example, the series {9, 5, 1, 0, 0, 0} has settled, or converged, on the number 0.

Integrals, limits, series and sequences can all converge. For example, if a limit settles on a certain (finite) number, then the limit exists. The opposite is diverge, where the integral, limit, series or sequence fails to settle on a number. In the case of a limit, if it diverges, then it doesn’t exist.

What Does Diverge Mean?

“Diverge” generally means either:

  • Settles on a certain number (i.e. has a limit), or
  • Doesn’t converge.

In some areas of math, diverge might simply mean “takes a different path” (for example, in KL Divergence in statistics). However, in calculus, it almost always pertains to limits or behavior of sequences and series.

Series and Sequences that Diverge (The Divergence Test)

divergence test
The divergence test.

Series and Sequences can also diverge. In a general sense, diverge means that the sequence or series doesn’t settle on a particular number.

A divergent series will (usually) go on and on to infinity (i.e. these series don’t have limits). For example, the series

9 + 11 + 13 …

will keep on growing forever.

Not all series diverge though: some diverge all the time, others converge or diverge under very specific circumstances. For example:

Proving divergence (or convergence) is extremely challenging with a few exceptions. For example, you can show that an infinite series diverges by showing that a sequence of partial sums diverges.

diverge or converge tests
Comparison of four popular tests (Boardman & Nleson, 2015).

Series Convergence Tests: Related Articles

Weierstrass M-Test

Series Convergence Tests: References

Arfken, G. (1985). Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press.
Boas, R. et al. (1996). A Primer of Real Functions. Cambridge University Press.
Browder, A. (1996). Mathematical Analysis: An Introduction. New York: Springer-Verlag, 1996.
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:
Hunter, K. Sequences and Series of Functions.
Jeffreys, H. and Jeffreys, B. S. (1988). “Uniform Convergence of Sequences and Series” et seq. §1.112-1.1155 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 37-43, 1988.
Kadak, U. (2014). On Uniform Convergence of Sequences and Series of Fuzzy-Valued Functions. Retrieved February 10, 2020 from:
Kevrekidis, P. 132class13 (PDF). Retrieved December 14, 2018 from:
Knopp, K. “Uniform Convergence.” §18 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 71-73, 1996.
Kuratowski, K. (2014). Introduction to Calculus. Elsevier.
Mathonline. Dirichlet’s Test for Convergence of Series of Real Numbers Examples 1. Retrieved September 18, 2020 from:
Nelson, D. (2008). The Penguin Dictionary of Mathematics. Penguin Books Limited.
Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, pp. 147-148.
Spivak, M. (2006). Calculus, 3rd edition. Cambridge University Press.
Vasishtha, A. Algebra & Trigonometry.
Vogel, T. Pointwise and Uniform Convergence of Sequences of Functions (7.1). Retrieved February 10, 2020 from:
Wood, A. (2012) Absolute and Conditional Convergence. Retrieved December 14, 2018 from:
Infographic based on Professor Joe Kahlig’s original graphic.

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