Root test

This article describes a test that is used to determine, in some cases, whether a given infinite series or improper integral converges. It may help determine whether we have absolute convergence, conditional convergence, or neither.
View a complete list of convergence tests

Statement

In limit superior form

Consider a series of the form:

$\sum _{k=1}^{\infty }a_{k}$

where all the $a_{k}$ are real numbers.

Note: It does not really matter where $k$ starts from. Some series are conventionally labeled to start from 0 and some to start from 1. If $k$ is starting from 0, we can simply ignore $a_{0}$.

The root test states the following: Consider the limit superior:

$L=\lim \sup _{k\to \infty }|a_{k}|^{1/k}$

allowing it to take the value $\infty$.

The following is true:

• If $L>1$ (we can include here the case $L=\infty$), the series diverges. In fact, the terms themselves do not approach zero, so no rearrangement of the series converges.
• If $L<1$ (this includes the case $L=0$, which is an extreme), the series converges. In fact, it is an absolutely convergent series: every rearrangement of the series converges to the same value.
• If $L=1$, the series may converge or diverge. Specifically, in this case, the test can be refined a little further:
  • If there are infinitely many values $|a_{k}|^{1/k}$ that are greater than or equal to 1, the terms $a_{k}$ do not go to zero, so the series diverges.
  • If there are only finitely many (and possibly no) values $|a_{k}|^{1/k}$ that are greater than or equal to 1, then the root test is genuinely inconclusive.

Replacing limit superior by limit

The root test can also be stated using the ordinary notion of limit instead of limit superior, i.e., by defining:

$L=\lim _{k\to \infty }|a_{k}|^{1/k}$

The rules are similar, because whenever the limit exists, it must equal the limit superior. However, this formulation of the test is weaker than the limit superior formulation. The key difference is that whereas the limit superior must always exist (either as a finite number or as $+\infty$) whereas the limit need not exist if the $a_{k}$s are oscillating. Thus, there are situations where the limit superior version is conclusive but the limit version isn't.

Explicitly, we need to make the following cases:

• If $L>1$ (we can include here the case $L=\infty$), the series diverges. In fact, the terms themselves do not approach zero, so no rearrangement of the series converges.
• If $L<1$ (this includes the case $L=0$, which is an extreme), the series converges. In fact, it is an absolutely convergent series: every rearrangement of the series converges to the same value.
• If $L=1$, the series may converge or diverge. Specifically, in this case, the test can be refined a little further:
  • If there are infinitely many values $|a_{k}|^{1/k}$ that are greater than or equal to 1, the terms $a_{k}$ do not go to zero, so the series diverges.
  • If there are only finitely many (and possibly no) values $|a_{k}|^{1/k}$ that are greater than or equal to 1, then the root test is genuinely inconclusive.
• If $L$ is not defined (i.e., the sequence $|a_{k}|^{1/k}$ does not have a finite limit or a limit of $+\infty$) the test is inconclusive. In this case, the test can be refined further as follows (the refinement is not part of the test per se):
  • If there are infinitely many values $|a_{k}|^{1/k}$ that are greater than or equal to 1, the terms $a_{k}$ do not go to zero, so the series diverges.
  • If there exists $c<1$ such that $|a_{k}|^{1/k}<c$ for all but finitely many $k$, then the series is an absolutely convergent series.
  • If all but finitely many of the $|a_{k}|^{1/k}$ are less than 1 but we cannot find $c<1$ that bounds all of them, then the root test is inconclusive.

Hierarchy of functions

List of functions

The big-oh notation $O(f(x))$ means a function that is bounded between constant positive multiples of $f(x)$ for $x$ large enough.

The following are functions with superexponential growth (which becomes superexponential decay when they appear in the denominator). They are arranged in decreasing order of growth:

1. Double exponential functions, i.e., functions of the form $a^{b^{x}}$ where $a,b>1$.
2. Function of the form $\exp(O(x^{r}))$, where $r>1$, i.e., functions growing exponentially in a superlinear power of $x$. Within these, the larger the value of $r$, the faster the growth.
3. Functions of the form $\exp(O(x\ln x))$. The factorial function lives here.

The only functions with exponential growth are functions of the form $\exp(kx)$, $k>0$. Their reciprocals, which are functions of the form $\exp(-kx),k>0$, give examples of functions of exponential decay.

Here are functions of subexponential growth (their reciprocals have subexponential decay):

1. Functions of the form $\exp(O(x^{r}))$, $0<r<1$
2. Functions that grow like polynomials or power functions, i.e., $O(x^{r}),r>0$. The larger the value of $r$, the faster the growth.
3. Functions that grow like polylogarithmic functions, i.e., $O((\ln x)^{r}),r>0$.

Interaction between functions

• Interaction between superexponentials: The product of two functions with superexponential growth still has superexponential growth. For the quotient of two functions with superexponential growth, first figure out which one is higher in the hierarchy. If the numerator is higher in the hierarchy, then we get superexponential growth overall. If the denominator is higher in the hierarchy, then we get superexponential decay overall. if they are at the same place in the hierarchy, try doing algebraic simplification and rewrite and see what you get -- you may end up getting anything.
• Interaction between superexponential and exponential/subexponential: The superexponential calls the shots.
• Interaction between exponential and subexponential: We still get something exponential, but the subexponential part is relevant to determining endpoint behavior.

Remember that all these are functions in $k$

Before we go on to apply the ideas of this hierarchy of functions to the convergence problem of power series, we need to remind ourselves of the following: we are interested in the growth or decay of the power series coefficients $a_{k}$, which are functions of the indexing variable $k$, not $x$. Thus, wherever, we see $x$ in the above hierarchy, we need to replace it by $k$.

Application of root test

The application of the root test is fairly straightforward:

Case on $|a_{k}|$ as a function of $k$	The limit superior of $|a_{k}|^{1/k}$	Examples	Conclusion
superexponential growth	$\infty$	$\sum _{k=0}^{\infty }2^{3^{k}}$, $\sum _{k=0}^{\infty }7^{k^{2}}$	diverges
superexponential decay	0	$\sum _{k=0}^{\infty }{\frac {1}{2^{4^{k}}}}$, $\sum _{k=0}^{\infty }{\frac {1}{2^{k^{5}}}}$	converges
exponential growth	strictly between 1 and $\infty$	$\sum _{k=3}^{\infty }6^{k}$	diverges
exponential decay	strictly between 0 and 1	$\sum _{k=1}^{\infty }{\frac {1}{2^{k}}}$	converges
subexponential growth	1	$\sum _{k=0}^{\infty }(k^{2}+1)x^{k}$	diverges, though this does not follow directly from the root test (rather, we see it from the fact that the terms do not approach zero)
subexponential decay	1	$\sum _{k=0}^{\infty }{\frac {x^{k}}{k+3}}$, $\sum _{k=0}^{\infty }{\frac {x^{k}}{k^{2}+4}}$	inconclusive

Relation with other tests

Ratio test

The ratio test is a considerably weaker test than the root test that only looks at ratios of successive terms and studies the limit of these ratios. It is more likely to be inconclusive. The main benefit of the ratio test is that it is easier to use in some circumstances.

Degree difference test

The standard example of a situation where the root test fails is a situation where the coefficients $a_{k}$ decay subexponentially, i.e., $a_{k}\to 0$ but $|a_{k}|^{1/k}\to 1^{-}$. Many examples of this kind fall within the purview of the degree difference test, which in turn is derived from the integral test.

Integral test

The integral test is an extremely sensitive test that can help determine convergence of series that are inconclusive both from the perspective of the ratio test and the degree difference test. These are series where the coefficients are decaying just slightly faster than the reciprocal of a linear function.

Proof

The root test follows essentially from a limit comparison test with a suitably constructed geometric series.