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## PowerPoint Slideshow about ' Power Series' - gaurav

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Presentation Transcript

Recall the Taylor polynomial of degree

for a function

times differentiable at

which is

Taylor PolynomialsSequence of Partial Sums

Each member of the sequence of

partial sums is itself a Taylor

polynomial.

A Taylor series is a

sequence of Taylor polynomials.

you need derivatives up to order

AssumptionsFor a Taylor polynomial,

For a Taylor series,

you need derivatives of all orders.

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

term

at

derivative of

Start by finding the

Now What?in the Taylor series for

Use the Calculator to...

get successive expressions for

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

at

A Weird ExampleIt’s hard to see that it even exists, but

Thus the coefficients in the Taylor series for

Meaning...

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.

Meaning...

- 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

ExampleFind a polynomial approximation to

Solution?

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.

So...

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.

means we can substitute

True For All x...That gives us

which is the Maclaurin series for that function.

More Cute Tricks

means

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

Arithmetic

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