Calculus How To

Series Convergence Tests

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Contents (Click to skip to that section):

  1. Series Convergence Tests in Alphabetical Order
  2. What Does “Converge” 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
converges.

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

Dirichlet’s Test

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


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


converges.

If L>1, then the series


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

Root Test

Let L = lim(n→ ∞)|an|1/n
If <, then the series


converges.
If >, then the series diverges.
If L=1, then the ratio test is inconclusive.

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.

Rate of Convergence

Rate of convergence tells you how fast a sequence of real numbers converges (reaches) a certain point or limit. It’s used as a tool to compare the speed of algorithms, particularly when using iterative methods.

Many different methods exist for calculating the rate of convergence. One straightforward way is with the following formula (Senning, 2020; Hundley, 2020),
rate of convergence

Where:

  • α = the order of convergence (a real number > 0) of the sequence. For example: 1 (linear), 2 (quadratic) or 3(cubic),
  • xn = a sequence,
  • λ = asymptotic error; A real number ≥ 1,
  • r = the value the sequence converges to.

In general, algorithms with a higher order of convergence reach their goal more quickly and require fewer iterations.

Radius and Interval of Convergence

A radius of convergence is associated with a power series, which will only converge for certain x-values. The interval where this convergence happens is called the interval of convergence, and is denoted by (-R, R). The letter R in this interval is called the radius of convergence. It’s called a “radius” because if the coefficients are complex numbers, the values of x (if |x| < R) will form an open disk of radius R.

Absolutely & Conditionally Convergent

Although you can generally say that something converges if it settles on a number, convergence in calculus is usually defined more strictly, depending on whether the convergence is conditional or absolute.

converge

An example of a geometric series. This one is absolutely convergent.



A series is absolutely convergent if the series converges and it also converges when all terms in the series are replaced by their absolute values.

Conditional Convergence is a special kind of convergence where a series is convergent when seen as a whole, but the absolute values diverge. It’s sometimes called semi-convergent.

A series is absolutely convergent if the series converges (approaches a certain number) and it also converges when all terms in the series are replaced by their absolute values. In other words,

…if |u1| + |u2| +… is convergent, then the series u1 + u2 +… is absolutely convergent.

This statement is usually written with the summation symbol:

if Σ |un| is convergent, then the series Σ un has absolute convergence.

Positive Terms Series

If the series of positive terms converges, then both the series of positive terms and the alternating series (i.e. a series with alternating positive and negative terms) will converge.

If a convergent series is a set of positive terms, then that series is also absolutely convergent. That’s because Σun and Σ|un| are the same series.
For example, the following geometric series is both:

absolutely convergent

A geometric series.


Series with Positive and Negative Terms

If a convergent series has an infinite number of positive terms and an infinite number of negative terms, it only has absolute convergence if Σ|un is also convergent.


Non-Absolute (Conditional) Convergence

A series is non-absolutely (conditionally) convergent if the series is convergent, but the set of absolute values for the series diverges. This is also called semi-convergence, or conditional convergence. For example, the following alternating series converges:


However, the series,

diverges.

choosing a series convergence test

Choosing a series convergence test for positive series.

Series Convergence Tests: Related Articles

Weierstrass M-Test


Series Convergence Tests: References

Hundley, D. Notes: Rate of Convergence. Retrieved September 8, 2020 from: http://people.whitman.edu/~hundledr/courses/M467F06/ConvAndError.pdf
Kevrekidis, P. 132class13 (PDF). Retrieved December 14, 2018 from: http://people.math.umass.edu/~kevrekid/132_f10/132class13.pdf
Kuratowski, K. (2014). Introduction to Calculus. Elsevier.
Senning, J. Computing and Estimating the Rate of Convergence. Retrieved September 8, 2020 from: http://www.math-cs.gordon.edu/courses/ma342/handouts/rate.pdf
Spivak, M. (2006). Calculus, 3rd edition. Cambridge University Press.
Vasishtha, A. Algebra & Trigonometry.
Wood, A. (2012) Absolute and Conditional Convergence. Retrieved December 14, 2018 from: https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2012/09/MA102-5.4.6-Absolute-and-Conditional-Convergence.pdf
Infographic based on Professor Joe Kahlig’s original graphic.

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Stephanie Glen. "Series Convergence Tests" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/sequence-and-series/series-convergence-tests/
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