Calculus II
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Ratio test and Root test

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01 Theory

Theory 1

Ratio Test (RaT)

Applicability: Any series with nonzero terms.

Test Statement:

Suppose that as .

Then:

Extra - Ratio Test - a deeper look

To understand the ratio test better, first consider this series:

  • The term is created by multiplying the prior term by .
  • The term is created 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.)

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02 Illustration

Example - Ratio test

Ratio test examples

(a) Observe that has ratio and thus . Therefore the RaT implies that this series converges.

Simplify the ratio:

Notice this technique! We frequently use these rules:

(To simplify ratios with 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 -series.

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03 Theory

Theory 2

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 . Set (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:

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04 Illustration

Example - 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.

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Example - 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 , 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.

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Series tests: strategy tips

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05 Theory

Theory 1

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

Notice large behavior is like or . This converges. Use the LCT to show absolute convergence. Skip the AST because absolute convergence settles the matter. Lesson: check for absolute BEFORE applying AST, even when alternating!

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).

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Power series: Radius and Interval

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06 Theory

Theory 1

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 becomes a variable , and each term has an additional coefficient parameter controlling the relative contribution of different orders.

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07 Theory

Theory 2

Every power series has a radius of convergence and an interval of convergence.

Radius of convergence

Consider a power series centered at :

Apply the ratio test:

Define the radius of convergence :

Therefore:

We can build shifted power series for near some other value . Just replace the variable with a shifted variable :

Now apply the ratio test to determine convergence:

Define the radius of convergence :

In the shifted setting, the radius of convergence limits the *distance from *:

Method: To calculate the interval of convergence of a power series, follow these steps:

  • Observe the center of the shifted series (or for no shift).
  • Compute using the limit of coefficient ratios.
  • Write down the preliminary interval .
  • Plug each endpoint, and , into the original series
    • Check for convergence
  • Add in the convergent endpoints. (4 possible scenarios.)
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08 Illustration

Example - Radius of convergence

Radius of convergence

Find the radius of convergence of the series:

(a) (b)

Solution

(a) Ratio of terms:

This converges for , or . Therefore .

(b) This power series skips the odd powers. Apply the ratio test to just the even powers:

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Example - Interval of convergence

Interval of convergence

Find the radius and interval of convergence of the following series:

(a) (b)

Solution

(a)

(1) Apply ratio test:

Therefore and thus:

Preliminary interval: .

(2) Check endpoints:

Check endpoint :

Check endpoint :

Final interval of convergence:

(b)

(1) Apply ratio test:

Therefore:

Preliminary interval:

(2) Check endpoints:

Check endpoint :

Check endpoint :

Final interval of convergence:

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Exercise - Radius and interval

Radius and interval for a few series

Find the radius and interval of convergence of the following series:

(a) (b)

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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 terms:

This converges when or .

Therefore and the preliminary interval is .

Check endpoints: diverges and also diverges.

Final interval is .

(b)

Ratio of terms:

This converges when or:

Preliminary interval:

Check endpoints: converges but diverges.

Final interval of convergence:

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