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Power Series. Dr. Dillon Calculus II Fall 1999. Recall the Taylor polynomial of degree. for a function. times differentiable at. which is. Taylor Polynomials. The Taylor series for a function. which has derivatives of all orders at a point. Definition. is given by. Compare.

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Power Series

Dr. Dillon

Calculus II

Fall 1999

Recall the Taylor polynomial of degree

for a function

times differentiable at

which is

Taylor Polynomials

The Taylor series for a function

which has derivatives of all orders at a point


is given by


Taylor polynomial

Taylor series

A Taylor Polynomial

is a (finite) sum

with a defined degree.

A Taylor Series

is an infinite sum, i.e.,

a sequence of partial sums.

Sequence of Partial Sums

Each member of the sequence of

partial sums is itself a Taylor


A Taylor series is a

sequence of Taylor polynomials.


Sequence of Taylor Polynomials


to be the sequence

As the Taylor Series

We actually define the Taylor series for


An Example

The first few Taylor polynomials for

The Taylor series:

you need derivatives up to order


For a Taylor polynomial,

For a Taylor series,

you need derivatives of all orders.






We could find Taylor series for


for the Taylor series for


Using sigma notation write


The Taylor series for

is called the Maclaurin series for



Will the Calculator Find a Taylor Series?

Maybe; try it for

Where Does One Start?

Start with a Taylor polynomial.

Try using a large degree.

Can you see the pattern?

I can’t!

Find the coefficient of the degree



derivative of

Start by finding the

Now What?

in the Taylor series for

Use the Calculator to...

get successive expressions for

The Pattern?

Look at a couple more:


is good for

and then



The coefficient for the degree

term is



in the Taylor series for


Finding a Taylor series

means finding the coefficients.

Taylor Series

Degree n term

Coefficient of Degree n Term

Separate Issues

  • What is the Taylor series for a function at a point?

  • For what values of x does the Taylor series for a function converge?

  • Does the Taylor series for a function converge to that function?


are all


A Weird Example

It’s hard to see that it even exists, but

Thus the coefficients in the Taylor series for


This Taylor series describes the function well

but only at one point, 0.

In cases like this, Taylor series aren’t good for much.

Lucky for Us

All of the usual suspects can be well represented by their Taylor series at all points where they are infinitely differentiable.

The Taylor series for all of our favorite functions converge to the functions at least on a decent sized interval, if not on the entire real line.


  • For all algebraic functions,

  • for trigonometric functions and their inverses,

  • for exponential functions and logarithms,

    there are excellent polynomial approximations to the functions on intervals surrounding points

where the functions have derivatives

of all orders.

that is to within two decimal places of

on the interval (2,6).

Use a Taylor polynomial at


Find a polynomial approximation to


What Degree Do We Need?

Use your technology to figure it out!

How Can You Tell?

If the polynomial and the function

agree at the endpoints,

they agree at all the points between.

That’s a Big Theorem.

A power series at

is a function of the form


Power Series

are the coefficients.

is a power series at


All the coefficients are 1.



for some positive number


Huge Theorem

is the Taylor series for f at a.


is the Maclaurin series for

More Interesting Stuff

If a power series converges on an interval, we can

  • differentiate term by term to get another convergent power series

  • integrate term by term to get another convergent power series

  • take limits term by term, on the interval of convergence

  • do arithmetic term by term to get still more convergent power series.

converges to


In other words,

for a given value of x


means we can substitute

True For All x...

That gives us

which is the Maclaurin series for that function.

More Cute Tricks


Find the Maclaurin series for

Start with the Maclaurin series for

It converges to

One Last Example


This is the Maclaurin series for


It converges to the function on the whole real line.

A Final Note

Memorize the following series:

  • The Maclaurin series for sine and cosine

  • The Maclaurin series for the natural exponential

The Maclaurin series for 1/(1-x)

Be Able to Use Them

to find Taylor series for

functions obtained from the above via

  • Differentiation

  • Integration


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