Calculus II
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Series tests: strategy tips

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01 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

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).

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Power series: Radius and Interval

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02 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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03 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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04 Illustration

Example - Radius of convergence

Radius of convergence

Find the radius of convergence of the series:

(a) (b)

Solution

(a) Ratio of terms:

Therefore and the series converges for .

(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:

Therefore and the preliminary interval is .

Check endpoints: diverges and also diverges.

Final interval is .

(b)

Ratio of terms:

Therefore:

Preliminary interval:

Check endpoints: converges but diverges.

Final interval of convergence:

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Power series as functions

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05 Theory

Theory 1

Given a numerical value for within the interval of convergence of a power series, the series sum may be considered as the output of a function .

Many techniques from algebra and calculus can be applied to such power series functions.

Addition and Subtraction:

Summation notation:

Scaling:

Summation notation:

Extra - Multiplication and composition

Multiplication:

For example, suppose that the geometric power series converges, so . Then we have for its square:

Composition:

Assume:

Then:

Differentiation:

Antidifferentiation:

For example, for the geometric series we have:

Do the series created with sums, products, derivatives etc., all converge? On what interval?

For the algebraic operations, the resulting power series will converge wherever both of the original series converge.

For calculus operations, the radius is preserved, but the endpoints are not necessarily:

Power series calculus - Radius preserved

If the power series has radius of convergence , then the power series and also have the same radius of convergence .

Power series calculus - Endpoints not preserved

It is possible that a power series converges at an endpoint of its interval of convergence, yet and do not converge at .

Extra - Proof of radius for derivative and integral series

Suppose has radius of convergence :

Consider now the derivative and its successive-term ratios:

Consider now the antiderivative and its successive-term ratios:

In both these cases the ratio test provides that the series converges when .

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06 Illustration

Example - Geometric series: algebra meets calculus

Geometric series: algebra meets calculus

Consider the geometric series as a power series functions:

Take the derivative of both sides of the function:

This means satisfies the identity:

Now compute the derivative of the series:

On the other hand, compute the square of the series:

So we find that the same relationship holds, namely , for the closed formula and the series formula for this function.

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Example - Manipulating geometric series: algebra

Manipulating geometric series: algebra

Find power series that represent the following functions:

(a) (b) (c) (d)

Solution

(a)

Rewrite in format :

Choose . Plug into geometric series:

Therefore:

(b)

Rewrite in format :

Choose . Plug into geometric series:

Therefore:

(c)

Rewrite in format :

Choose . Here . Plug into geometric series:

Therefore:

(d)

Rewrite in format :

Choose . Here . Plug into geometric series:

Therefore:

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Example - Manipulating geometric series: calculus

Manipulating geometric series: calculus

Find a power series that represents .

Solution

Differentiate to obtain similarity to geometric sum formula:

Integrate series to find original function:

Use known point to solve for :

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Example - Recognizing and manipulating geometric series: Part I

Recognizing and manipulating geometric series: Part I

(a) Evaluate . (Hint: consider the series of .)

(b) Find a series approximation for .

Solution

(a)

(1) Follow hint, study series of :

Notice:

Integrate the series:

Solve for using which (plugging above) implies and thus . So:

(2) Relate to the given series:

Notice that if we set . Also, . Therefore:

So the answer is .

(b) Find a series approximation for :

Observe that .

Plug into the series:

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Example - Recognizing and manipulating geometric series: Part II

Recognizing and manipulating geometric series: Part II

(a) Find a series representing using differentiation.

(b) Find a series representing .

Solution

(a)

Notice that .

What is the series for ?

Let :

Now integrate this by terms:

Conclude:

Plug in to solve for :

Final answer:

(b)

Rewrite integrand in format of geometric series sum:

Therefore:

Integrate the series by terms to obtain the answer:

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