Ratio test and Root test
Videos, Math Dr. Bob
- Ratio test: Basics
- Ratio test: Ratio test + DCT
- Root test: Basics
- Root test: for
01 Theory
Ratio Test (RaT)
Applicability: Any series with nonzero terms.
Test Statement:
Suppose that
as . Then:
Extra - Ratio test: explanation
To understand the ratio test, consider this series:
- The term
is given by multiplying the prior term by . - The term
is given by multiplying the prior term by . - The term
is created by multiplying the prior term by . When
, the multiplication factor giving the next term is necessarily less than . Therefore, when , the terms shrink faster than those of a geometric series having . Therefore this series converges. Similarly, consider this series:
Write
for the ratio from the prior term to the current term . For this series, . This ratio falls below
when , after which the terms necessarily shrink faster than those of a geometric series with . Therefore this series converges. The main point of the discussion can be stated like this:
Whenever this is the case, then eventually the ratios are bounded below some
, and the series terms are smaller than those of a converging geometric series.
Extra - Ratio test: proof
Let us write
for the ratio to the next term from term . Suppose that
as , and that . This means: eventually the ratio of terms is close to ; so eventually it is less than . More specifically, let us define
. This is the point halfway between and . Since , we know that eventually . Any geometric series with ratio
converges. Set for big enough that . Then the terms of our series satisfy , and the series starting from is absolutely convergent by comparison to this geometric series. (Note that the terms
do not affect convergence.)
02 Illustration
Example - Ratio test
Ratio test examples
(a) Observe that
has ratio and thus . Therefore the RaT implies that this series converges. Notice this technique!
Simplify the ratio:
We frequently use these rules:
to simplify ratios having exponents and factorials.
(b)
has ratio . Simplify this:
So the series converges absolutely by the ratio test.
(c) Observe that
has ratio as . So the ratio test is inconclusive, even though this series fails the SDT and obviously diverges.
(d) Observe that
has ratio as . So the ratio test is inconclusive, even though the series converges as a
-series with . (e) More generally, the ratio test is usually inconclusive for rational functions; it is more effective to use LCT with a
Link to original-series.
03 Theory
Root Test (RooT)
Applicability: Any series.
Test Statement:
Suppose that
as . Then:
Extra - Root test: explanation
The fact that
and implies that eventually for all high enough , where is the midpoint between and . Now, the equation
is equivalent to the equation . Therefore, eventually the terms
are each less than the corresponding terms of this convergent geometric series:
04 Illustration
Root test examples
Root test examples
(a) Observe that
has roots of terms: Because
, the RooT shows that the series converges absolutely. (b) Observe that
has roots of terms: Because
, the RooT shows that the series converges absolutely. (c) Observe that
Link to originalconverges because as .
Ratio test versus root test
Ratio test versus root test
Determine whether the series
converges absolutely or conditionally or diverges. Solution Before proceeding, rewrite somewhat the general term as
. Now we solve the problem first using the ratio test. By plugging in
we see that So for the ratio
we have: Therefore the series converges absolutely by the ratio test.
Now solve the problem again using the root test. We have for
: To compute the limit as
we must use logarithmic limits and L’Hopital’s Rule. So, first take the log: Then for the first term apply L’Hopital’s Rule:
So the first term goes to zero, and the second (constant) term is the value of the limit. So the log limit is
Link to original, and the limit (before taking logs) must be (inverting the log using ) and this is . Since , the root test also shows that the series converges absolutely.
Series tests: strategy tips
Videos, Math Dr. Bob
- Series test round-up: Part I
- Series test round-up: Part II
- Series test round-up: Part III
Videos, Trefor Bazett
05 Theory
It can help to associate certain “strategy tips” to find convergence tests based on certain patterns.
Matching powers → Simple Divergence Test
Use the SDT because we see the highest power is the same (
) in numerator and denominator.
Rational or Algebraic → Limit Comparison Test
Use the LCT because we have a rational or algebraic function (positive terms).
Not rational, not factorials → Integral Test
Use the IT because we do not have a rational/algebraic function, and we do not see factorials.
Rational, alternating → AST, and LCT or DCT
Use the AST because it’s alternating. Then use the LCT (to find absolute convergence) because its a rational function.
Factorials → Ratio Test
Use the RaT because we see a factorial. (In case of alternating + factorial, use RaT first.)
Recognize geometric → LCT or DCT
Use the LCT or DCT comparing to
because we see similarity to (recognize geometric).
Power series: Radius and Interval
Videos, Math Dr. Bob
- Power series: Interval and Radius of Convergence
- Power series: Interval of Convergence Using Ratio Test
- Power series: Interval of Convergence Using Root Test
- Power series: Finding the Center
06 Theory
A power series looks like this:
Power series are used to build and study functions. They allow a uniform “modeling framework” in which many functions can be described and compared. Power series are also convenient for computers because they provide a way to store and evaluate differentiable functions with numerical (approximate) values.
Small
needed for power series The most important fact about power series is that they work for small values of
. Many power series diverge for
too big; but even when they converge, for big they converge more slowly, and partial sum approximations are less accurate.
The idea of a power series is a modification of the idea of a geometric series in which the common ratio
07 Theory
Every power series has a radius of convergence and an interval of convergence.
Radius of convergence
Consider a power series centered at
: Define
as the limit of coefficient ratios: Then reciprocal,
, is the radius of convergence; it can be anything in including either extreme. The power series necessarily converges for
and diverges for .
Extra - Radius of convergence: explanatory proof
Treat the variable
in the power series as a constant. Apply the ratio test to this series. The ratio function is:
Since
is a constant here, we have: Therefore, the ratio test says that the series converges absolutely when
, and diverges when .
We can build shifted power series for
The radius of convergence of a shifted series is calculated in the same way, using the coefficients:
However, in the shifted setting, the radius of convergence concerns the distance from
The interval of convergence of a power series is determined by:
- the radius of convergence
- the center point
- special consideration of endpoints
Interval of convergence
The interval of convergence
of a power series is the set of values of where the series converges. The interval of convergence
is:
- centered at
- extending a distance
to either side of - including / excluding the endpoints where
depending on the particular case
To calculate the interval of convergence, follow these steps:
- Observe the center
of the shifted series; corresponds to no shift. - Take the limit to compute
. - Write down the preliminary interval
. - Plug each endpoint
and into the original series - → check for convergence
- Add in the convergent endpoints. There are 4 total possibilities.
08 Illustration
Example - Radius and interval for a few series
Radius and interval for a few series
Link to original
Series Radius Interval
Example - Radius of convergence
Radius of convergence
Find the radius of convergence of the series:
(a)
(b) Solution
(a) The ratio of coefficients is
. Therefore
and the series converges for . (b) This power series has
, meaning it skips all odd terms. Instead of the standard ratio function, we take the ratio of successive even terms. The series of even terms has coefficients
. So: As
Link to original, this converges to , so and .
Example - Interval of convergence
Interval of convergence
Find the interval of convergence of the following series.
(a)
(b) Solution
(a)
- Apply ratio test.
- Ratio of successive coefficients:
- Limit of ratios:
- Deduce
and therefore . - Therefore:
- Preliminary interval of convergence.
- Translate to interval notation:
- Final interval of convergence.
- Check endpoint
:
- Check endpoint
:
- Final interval of convergence:
(b)
Link to original
- Limit of coefficients ratio.
- Ratio of successive coefficients:
- Limit of ratios:
- Deduce
and thus . - Therefore:
- Preliminary interval of convergence:
- Check endpoints.
- Check endpoint
:
- Check endpoint
:
- Final interval of convergence:
Interval of convergence - further examples
Interval of convergence - further examples
Find the interval of convergence of the following series.
(a)
(b) Solution
(a)
- Ratio of coefficients:
. - So the
, center is , and the preliminary interval is . - Check endpoints:
diverges and also diverges. Final interval is . (b)
Link to original
- Ratio of coefficients:
. - So
, and the series converges when . - Extract preliminary interval.
- Divide by
: - Check endpoints:
converges but diverges. - Final interval of convergence: