Convergence tests

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$.

List of tests

Limit of the summand

If the limit of the summand is undefined or nonzero, that is ${\displaystyle \lim _{n\to \infty }a_{n}\neq 0}$, then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.

Ratio test

This is also known as d'Alembert's criterion.

Consider two limits ${\displaystyle \ell =\liminf _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|}$ and ${\displaystyle L=\limsup _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|}$. If ${\displaystyle \ell >1}$, the series diverges. If ${\displaystyle L<1}$ then the series converges absolutely. If ${\displaystyle \ell \leq 1\leq L}$ then the test is inconclusive, and the series may converge absolutely, conditionally or diverge.

Root test

This is also known as the nth root test or Cauchy's criterion.

Let

${\displaystyle r=\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}},}$

where ${\displaystyle \limsup }$ denotes the limit superior (possibly ${\displaystyle \infty }$; if the limit exists it is the same value).

If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.^[1]

Integral test

The series can be compared to an integral to establish convergence or divergence. Let ${\displaystyle f:[1,\infty )\to \mathbb {R} _{+}}$ be a non-negative and monotonically decreasing function such that ${\displaystyle f(n)=a_{n}}$. If $${\displaystyle \int _{1}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty ,}$$ then the series converges. But if the integral diverges, then the series does so as well. In other words, the series ${\displaystyle {a_{n}}}$ converges if and only if the integral converges.

p-series test

A commonly-used corollary of the integral test is the p-series test. Let ${\displaystyle k>0}$. Then ${\displaystyle \sum _{n=k}^{\infty }{\bigg (}{\frac {1}{n^{p}}}{\bigg )}}$ converges if ${\displaystyle p>1}$.

The case of ${\displaystyle p=1,k=1}$ yields the harmonic series, which diverges. The case of ${\displaystyle p=2,k=1}$ is the Basel problem and the series converges to ${\displaystyle {\frac {\pi ^{2}}{6}}}$. In general, for ${\displaystyle p>1,k=1}$, the series is equal to the Riemann zeta function applied to ${\displaystyle p}$, that is ${\displaystyle \zeta (p)}$.

Direct comparison test

If the series ${\displaystyle \sum _{n=1}^{\infty }b_{n}}$ is an absolutely convergent series and ${\displaystyle |a_{n}|\leq |b_{n}|}$ for sufficiently large n , then the series ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ converges absolutely.

Limit comparison test

If ${\displaystyle \{a_{n}\},\{b_{n}\}>0}$, (that is, each element of the two sequences is positive) and the limit ${\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}}$ exists, is finite and non-zero, then either both series converge or both series diverge.

Cauchy condensation test

Let ${\displaystyle \left\{a_{n}\right\}}$ be a non-negative non-increasing sequence. Then the sum ${\displaystyle A=\sum _{n=1}^{\infty }a_{n}}$ converges if and only if the sum ${\displaystyle A^{*}=\sum _{n=0}^{\infty }2^{n}a_{2^{n}}}$ converges. Moreover, if they converge, then ${\displaystyle A\leq A^{*}\leq 2A}$ holds.

Abel's test

Suppose the following statements are true:

1. ${\displaystyle \sum a_{n}}$ is a convergent series,
2. ${\displaystyle \left\{b_{n}\right\}}$ is a monotonic sequence, and
3. ${\displaystyle \left\{b_{n}\right\}}$ is bounded.

Then ${\displaystyle \sum a_{n}b_{n}}$ is also convergent.

Absolute convergence test

Every absolutely convergent series converges.

Alternating series test

Suppose the following statements are true:

• ${\displaystyle (a_{n})_{n=1}^{\infty }}$ is monotonic,
• ${\displaystyle \lim _{n\to \infty }a_{n}=0}$

Then ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}}$ and ${\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}a_{n}}$ are convergent series. This test is also known as the Leibniz criterion.

Dirichlet's test

If ${\displaystyle \{a_{n}\}}$ is a sequence of real numbers and ${\displaystyle \{b_{n}\}}$ a sequence of complex numbers satisfying

• ${\displaystyle a_{n}\geq a_{n+1}}$

• ${\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}$

• ${\displaystyle \left|\sum _{n=1}^{N}b_{n}\right|\leq M}$ for every positive integer N

where M is some constant, then the series

${\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}}$

converges.

Cauchy's convergence test

A series ${\displaystyle \sum _{i=0}^{\infty }a_{i}}$ is convergent if and only if for every ${\displaystyle \varepsilon >0}$ there is a natural number N such that

${\displaystyle |a_{n+1}+a_{n+2}+\cdots +a_{n+p}|<\varepsilon }$

holds for all n > N and all p ≥ 1.

Stolz–Cesàro theorem

Let ${\displaystyle (a_{n})_{n\geq 1}}$ and ${\displaystyle (b_{n})_{n\geq 1}}$ be two sequences of real numbers. Assume that ${\displaystyle (b_{n})_{n\geq 1}}$ is a strictly monotone and divergent sequence and the following limit exists:

${\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l.\ }$

Then, the limit

${\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l.\ }$

Weierstrass M-test

Suppose that (f_n) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (M_n) satisfying the conditions

• ${\displaystyle |f_{n}(x)|\leq M_{n}}$ for all ${\displaystyle n\geq 1}$ and all ${\displaystyle x\in A}$, and
• ${\displaystyle \sum _{n=1}^{\infty }M_{n}}$ converges.

Then the series

${\displaystyle \sum _{n=1}^{\infty }f_{n}(x)}$

converges absolutely and uniformly on A.

Extensions to the ratio test

The ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case.

Raabe–Duhamel's test

Let { a_n } be a sequence of positive numbers.

Define

${\displaystyle b_{n}=n\left({\frac {a_{n}}{a_{n+1}}}-1\right).}$

If

${\displaystyle L=\lim _{n\to \infty }b_{n}}$

exists there are three possibilities:

• if L > 1 the series converges (this includes the case L = ∞)
• if L < 1 the series diverges
• and if L = 1 the test is inconclusive.

An alternative formulation of this test is as follows. Let { a_n } be a series of real numbers. Then if b > 1 and K (a natural number) exist such that

${\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|\leq 1-{\frac {b}{n}}}$

for all n > K then the series {a_n} is convergent.

Bertrand's test

Let { a_n } be a sequence of positive numbers.

Define

${\displaystyle b_{n}=\ln n\left(n\left({\frac {a_{n}}{a_{n+1}}}-1\right)-1\right).}$

If

${\displaystyle L=\lim _{n\to \infty }b_{n}}$

exists, there are three possibilities:^[2][3]

• if L > 1 the series converges (this includes the case L = ∞)
• if L < 1 the series diverges
• and if L = 1 the test is inconclusive.

Gauss's test

Let { a_n } be a sequence of positive numbers. If ${\displaystyle {\frac {a_{n}}{a_{n+1}}}=1+{\frac {\alpha }{n}}+O(1/n^{\beta })}$ for some β > 1, then ${\displaystyle \sum a_{n}}$ converges if α > 1 and diverges if α ≤ 1.^[4]

Kummer's test

Let { a_n } be a sequence of positive numbers. Then:^[5][6][7]

(1) ${\displaystyle \sum a_{n}}$ converges if and only if there is a sequence ${\displaystyle b_{n}}$ of positive numbers and a real number c > 0 such that ${\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\geq c}$.

(2) ${\displaystyle \sum a_{n}}$ diverges if and only if there is a sequence ${\displaystyle b_{n}}$ of positive numbers such that ${\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\leq 0}$

and ${\displaystyle \sum 1/b_{n}}$ diverges.

Abu-Mostafa's test

Let ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ be an infinite series with real terms and let ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ be any real function such that ${\displaystyle f(1/n)=a_{n}}$ for all positive integers n and the second derivative ${\displaystyle f''}$ exists at ${\displaystyle x=0}$. Then ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ converges absolutely if ${\displaystyle f(0)=f'(0)=0}$ and diverges otherwise.^[8]

Notes

• For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

Examples

Consider the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{\alpha }}}.}$

(i)

Cauchy condensation test implies that (i) is finitely convergent if

${\displaystyle \sum _{n=1}^{\infty }2^{n}\left({\frac {1}{2^{n}}}\right)^{\alpha }}$

(ii)

is finitely convergent. Since

${\displaystyle \sum _{n=1}^{\infty }2^{n}\left({\frac {1}{2^{n}}}\right)^{\alpha }=\sum _{n=1}^{\infty }2^{n-n\alpha }=\sum _{n=1}^{\infty }2^{(1-\alpha )n}}$

(ii) is a geometric series with ratio ${\displaystyle 2^{(1-\alpha )}}$. (ii) is finitely convergent if its ratio is less than one (namely ${\displaystyle \alpha >1}$). Thus, (i) is finitely convergent if and only if ${\displaystyle \alpha >1}$.

Convergence of products

While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let ${\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }}$ be a sequence of positive numbers. Then the infinite product ${\displaystyle \prod _{n=1}^{\infty }(1+a_{n})}$ converges if and only if the series ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ converges. Also similarly, if ${\displaystyle 0<a_{n}<1}$ holds, then ${\displaystyle \prod _{n=1}^{\infty }(1-a_{n})}$ approaches a non-zero limit if and only if the series ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ converges .

This can be proved by taking the logarithm of the product and using limit comparison test.^[9]

See also

• L'Hôpital's rule
• Shift rule

References

1. Wachsmuth, Bert G. "MathCS.org - Real Analysis: Ratio Test". www.mathcs.org.
2. František Ďuriš, Infinite series: Convergence tests, pp. 24–9. Bachelor's thesis.
3. Weisstein, Eric W. "Bertrand's Test". mathworld.wolfram.com. Retrieved 2020-04-16.
4. • "Gauss criterion", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
5. "Über die Convergenz und Divergenz der unendlichen Reihen". Journal für die reine und angewandte Mathematik. 1835 (13): 171–184. 1835-01-01. doi:10.1515/crll.1835.13.171. ISSN 0075-4102. S2CID 121050774.
6. Tong, Jingcheng (1994). "Kummer's Test Gives Characterizations for Convergence or Divergence of all Positive Series". The American Mathematical Monthly. 101 (5): 450–452. doi:10.2307/2974907. JSTOR 2974907.
7. Samelson, Hans (1995). "More on Kummer's Test". The American Mathematical Monthly. 102 (9): 817–818. doi:10.1080/00029890.1995.12004667. ISSN 0002-9890.
8. Abu-Mostafa, Yaser (1984). "A Differentiation Test for Absolute Convergence" (PDF). Mathematics Magazine. 57 (4): 228–231.
9. Belk, Jim (26 January 2008). "Convergence of Infinite Products".

Further reading

• Leithold, Louis (1972). The Calculus, with Analytic Geometry (2nd ed.). New York: Harper & Row. pp. 655–737. ISBN 0-06-043959-9.