Absolute Convergence and Ratio and Root Tests

An infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite. More precisely, a real or complex series $\textstyle\sum_{n=0}^\infty a_n$ is said to converge absolutely if $\textstyle\sum_{n=0}^\infty \left|a_n\right| = L$ for some real number $L$. Similarly, an improper integral of a function, $\textstyle\int_0^\infty f(x)\,dx$, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\int_0^\infty \left|f(x)\right|dx = L$.

Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly. (A convergent series that is not absolutely convergent is called conditionally convergent.)

Ratio Test

The ratio test is a test (or "criterion") for the convergence of a series $\sum_{n=1}^\infty a_n$, where each term is a real or complex number and $a_n$ is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test.

The usual form of the test makes use of the limit, $L = \lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|$. The ratio test states that,

• if $L < 1$, then the series converges absolutely;
• if $L > 1$, then the series does not converge;
• if $L = 1$ or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

Root Test

The root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity $\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}$, where $a_n$ are the terms of the series, and states that the series converges absolutely if this quantity is less than one but diverges if it is greater than one. It is particularly useful in connection with power series.

The root test was developed first by Augustin-Louis Cauchy and so is sometimes known as the Cauchy root test, or Cauchy's radical test. For a series $\sum_{n=1}^\infty a_n$, the root test uses the number $C = \limsup_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}$, where "lim sup" denotes the limit superior, possibly ∞. Note that if $\lim_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}$ converges, then it equals $C$ and may be used in the root test instead. The root test states that

• if $C < 1$, then the series converges absolutely;
• if $C > 1$, then the series diverges;
• if $C = 1$ and the limit approaches strictly from above, then the series diverges;
• otherwise the test is inconclusive (the series may diverge, converge absolutely, or converge conditionally).

There are some series for which $C = 1$ and the series converges, e.g.:

$\displaystyle{\sum{\frac{1}{n^2}}}$

and there are others for which $C = 1$ and the series diverges, e.g.:

 $\displaystyle{\sum{\frac{1}{n}}}$

Ratio Test

In this example, the ratio of adjacent terms in the blue sequence converges to $L=\frac{1}{2}$. We choose $r = \frac{L+1}{2} = \frac{3}{4}$. Then the blue sequence is dominated by the red sequence for all $n \geq 2$. The red sequence converges, so the blue sequence does as well.