Taylor Series and Taylor

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Taylor Series and Taylors Theorem

• When is a function given by its Taylor Series?

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So where were we?

• Facts
• f is continuous and has derivatives of all
  orders at x 0.
• f (n)(0)0 for all n.

This tells us that the Maclaurin Series for f is
zero everywhere!
The Maclaurin Series for f converges everywhere,
but is equal to f only at x 0!
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This tells us that.
Our ability write down a Taylor series for a
function is not in itself a guarantee that the
series will any anything to do with the function,
even on its interval of convergence!
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However, Ostebee and Zorn assures that.
Taylors theorem guarantees that this
unfortunate event seldom occurs. In other
words, the functions that are not given by their
Taylor series are pretty weird. Most of our
everyday functions ARE given by their Taylor
Series.
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Recall Taylors Theorem
Suppose that f is repeatedly differentiable on an
interval I containing x0 and that is the nth
order Taylor polynomial based at x0. Suppose
that Kn1 is a number such that for all z in
I, Then for x in I, (Page 504 in OZ)
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Pinning this down

• Recall that Pn is the nth partial sum of
  theTaylor Series of f based at x0.
• And thus
• Measures the error made by Pn(x) in approximating
  f (x).
• Taylors theorem gives us an upper bound on this
  error!

The Taylor series for f will converge to f if
and only if for all x f (x) - Pn(x) goes to
zero as n ?8. Taylors theorem can help us
establish this.
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Using Taylors Theorem

1. Find the Taylor series for f that is based at x
   p/4.
2. Show that this Taylor series converges to f for
   all values of x.

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1. Taylor Series for f (x) sin(x)
n f (n)(x) f (n)( ) an f (n)( )/n!
0
1
2
3
4
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Show that this converges to sin(x)
We start with the general set-up for Taylors
Theorem. What is Kn1? It follows that for all
x
What happens to this quantity As n?8?
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Show that this converges to sin(x)
Notice that I didnt have to know what Pn was in
order to gather this information. (In other
words, our second question is independent of our
first.)
We start with the general set-up for Taylors
Theorem. What is Kn1? It follows that for all
x
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Now its your turn
Repeat this exercise with the Maclaurin series
for f (x) cos(2x) .

1. Find the Maclaurin series for f (x) cos(2x).
2. Show that this series converges to f for all
   values of x.

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1. Taylor Series for f (x) cos(2x)
n f (n)(x) f (n)(0) an f (n)(0)/n!
0
1
2
3
4
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Show that this converges to cos(2x)
We start with the general set-up for Taylors
Theorem. What is Kn1? It follows that for
all x,
This quantity goes to 0 as n?8!
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Epilogue---Two points of view
Power series as functions
Taylor Series
First a series . . .
First a function . . .
. . . Then a function
. . . Then a series
Guarantees that f is equal to the power series
where the power series converges.
No a priori guarantee that f is equal to its
Taylor series.
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Why the Taylor series, then?
Power series as functions
Taylor Series
First a series . . .
First a function . . .
. . . Then a function
. . . Then a series
If f is equal to any power series at all, that
power series must be the Taylor series for f.
Thats why thats were we look!
Guarantees that the power series we started with
is, in fact, the TAYLOR SERIES FOR f .