Taylor and Maclaurin Series

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Taylor and Maclaurin Series

• Taylor Polynomials at x 0
• Basic Taylor Polynomials
• Taylor Series and Maclaurin Series
• Taylor Polynomials at x a
• Finding Taylor Series by Substitutions,
  Differentiation and Integration

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Functions Represented by Power Series
Consider a function f represented by a
converging power series f(x) a0 a1x a2x2
a3x3 .
Clearly f(0) a0.
To compute the other coefficients ak,
differentiate term by term to get f(x) a1
2a2x .
Insert x 0 to the above to get f(0) a1.
Conclude
A converging power series representing a function
f is necessarily of the above form. This power
series is called Maclaurin Series, a special form
a more general Taylor Series.
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Approximating Functions
Another way to get to Taylor Series is to
consider approximations of functions by
polynomials.
We assume that the function f is has
derivatives of all orders everywhere in its
domain of definition.
The Taylor polynomial of
degree n for given
function f at a point a is a polynomial P
of degree n such that P(k)
(a)f (k)(a) for k0,1,,n. This means that
the value of the polynomial P and all of its
derivatives up to the order n agree with those
of the function f at the point xa.
Definition
Observe that the defining conditions for the
Taylor polynomial have to do with the behavior of
the polynomial at one point only.
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Taylor Polynomials at x 0
Straightforward differentiation yields.
The general formula is
We conclude that the Taylor Polynomial for an
infinitely differentiable function f at x0 is
uniquely defined, and that the coefficients ak
are given by the above formula.
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Explicit Taylor Polynomials
Formula
By the preceding considerations, the Taylor
Polynomial of degree n of a function f is the
polynomial Tf,n(x)

• Using this formula Taylor polynomials of
  functions can often be rather easily computed.
  Strategy is the following
• Compute several derivatives of the given
  function.
• Evaluate these derivatives at x 0.
• Detect a pattern to find a general formula for
  f(n)(0).

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Taylor Polynomials for the Sine Function
To find the Taylor polynomials of the function
f(x) sin(x) compute derivatives and evaluate
them at x 0. One gets
Conclude
All even order derivatives of sin take the value
0 at x 0. Odd order derivatives take the
values 1 and -1.
Conclusion as a Formula
Formula
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Taylor Polynomials for Cosine
To find the Taylor polynomials of the function
g(x) cos(x) compute derivatives and evaluate
them at x 0. One gets
Conclude
All odd order derivatives of cos take the value 0
at x 0. Even order derivatives take the
values 1 and -1.
Conclusion as a Formula
Formula
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Taylor Polynomials for the Exponential Function
To find the Taylor polynomials of the exponential
function h(x) ex compute derivatives and
evaluate them at x 0. One gets
Conclude
The value of the exponential function and that of
all of its derivatives at x 0 is 1.
Formula
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Taylor Polynomials for the Sine Function
Formula
The following figure illustrates Taylor
polynomials of degrees 5 (blue), 9 (red) and 15
(green) for the sine function.
One concludes from the picture that all of the
above Taylor approximations for the sine function
appear to approximate the function well near the
origin (center of the above picture). Higher
order Taylor polynomials approximate better away
from the origin.
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Basic Taylor Polynomials
1
2
3
We will later see that the above polynomials can
be used to approximate the values of the
respective functions for all x.
The Taylor Polynomials of the function f(x)
(1x)p are given below. They can be used to
approximate the values of the function only for
-1ltxlt1.
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Taylor Polynomials at x a
The conditions used to define a Taylor polynomial
P of a given function f require that the
polynomial P and all of its non-zero
derivatives at a point xa agree with those of
the function f. Clearly the definition implies
that the polynomial P approximates the function
f best near the point xa.
Formula for Taylor Polynomials at xa
Assume the function f has all derivatives at
the point xa.
Taylor polynomial of degree n at xa is
The above formula follows by directly computing
the values of the derivatives of the function f
and those of the polynomial P at xa.
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Goodness of Approximations
The following figure shows the graph of the sine
functions and those of its Taylor polynomials of
degree 5 at the points x-p, x0, xp, x2p,
x3p.
The Taylor polynomial of degree 5 at the point
x-p approximates the sine function so well near
the point x-p that its graphs is completely
covered by the black graph of the sine function
near that point. As xlt -3p/2 or xgt -p/2 the
approximation fails to follow the graph of the
sine function. These portions of the graph of
the Taylor polynomial of degree 5 at x-p are
shown as the left most red graphs above and under
the x -axis.
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Taylor Series and Maclaurin series
Letting n grow the Taylor polynomials at x a
define Taylor series at x a for the respective
functions. Basic Taylor Polynomials yield the
following Basic Taylor Series at x 0. The
Taylor Series at x 0 are also called Maclaurin
series.
These series converge and represent the given
function for all x. This will be shown later.
The Binomial Series
This is valid for -1ltxlt1.
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Taylor Polynomial Approximations
Problem
Solution
The error is given by
The Taylor series for sin(x) is an alternating
series.
The error done when approximating sin(x) by a
Taylor polynomial of degree 5 is bounded by the
absolute value of the first term left out. Hence
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Binomial Series and Geometric Series
The Binomial Series
Insert p -1 to the above to get
The binomial series for p -1, is the geometric
series with the first term 1 and with q x as
the ratio of two subsequent terms.
Conclude
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Binomial Series Example
Problem
Find Taylor series for the function f(x) 1/(1
- x)2.
Use the Binomial Series
Insert p -2 to the above to get
Alternative derivation for this series expansion
follows later.
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Finding Taylor Series

• One can find Taylor series for complicated
  functions by
• Substitutions
• Integrating a known series term by term
• Differentiating a known series term by term
• Any combination of the above tricks

One usually starts with one of the basic Taylor
series and manipulates that to get the desired
Taylor series. The above tricks are legal
provided that the series in question converge and
represent the functions in question. This
depends on the function for which Taylor series
representation needs to be derived. Many of the
basic Taylor series converge everywhere.
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Finding Taylor Series by Substitution
Problem
Solution
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Finding Taylor Series by Integration
Problem
Solution
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Taylor Series for ln(1 x)
Formula
This figure shows the graph of the function ln(1
x) and those of its Taylor polynomials of order
4 (blue), 9 (red), 14 (green) and 19 (yellow).
One observes that up to x 1 higher order
Taylor polynomials give better approximations
than lower order Taylor polynomials. For x gt 1,
the situation is reversed the higher the order,
the worse the approximation. This reflects the
fact that the Taylor series for ln(1 x) does
not converge for x gt 1.
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Finding Taylor Series by Differentiation
Problem
Solution
The above formula is a special case of the
binomial series and it converges for x lt 1.