TAYLOR SERIES

This slide show consists of graphs of various functions together with some of
their Taylor polynomials about the origin.

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A.  exp(x)
     This graphs exp(x) in the interval -2 < x < 2, and then overlays it with
the polynomials
     1
     1 + x
     1 + x + x^2/2!
     1 + x + x^2/2! + x^3/3!
     1 + x + x^2/2! + x^3/3! + x^4/4!
     1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5!
The radius of convergence of the Taylor series is infinity.

B.  sin(x)
     This graphs sine(x) in the interval 0 < x < 2ã, and then overlays it with
the polynomials
     x
     x - x^3/3!
     x - x^3/3! + x^5/5!
     x - x^3/3! + x^5/5! - x^7/7!
     x - x^3/3! + x^5/5! - x^7/7! + x^9/9!
     x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - x^11/11!
The radius of convergence of the Taylor series is infinity.

C.  cos(x)
     This graphs cosine(x) in the interval 0 < x < 2ã, and then overlays it with
the polynomials
     1
     1 - x^2/2!
     1 - x^2/2! + x^4/4!
     1 - x^2/2! + x^4/4! - x^6/6!
     1 - x^2/2! + x^4/4! - x^6/6! + x^8/8!
     1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - x^10/10!
The radius of convergence of the Taylor series is infinity.

D.  1/(1 - x), -1 < x < 1
     This graphs 1/(1 - x) in the interval -1 < x < 1, and then overlays it with
the polynomials
     1
     1 + x
     1 + x + x^2
     1 + x + x^2 + x^3
     1 + x + x^2 + x^3 + x^4
     1 + x + x^2 + x^3 + x^4 + x^5
The radius of convergence of the Taylor series is 1.  Notice what is happening
near -1.  The larger the polynomial, the better it is at approximating the
function.

E.  1/(1 - x), -2 < x < 2
     This graphs 1/(1 - x) in the interval -2 < x < 2, and then overlays it with
the polynomials
     1
     1 + x
     1 + x + x^2
     1 + x + x^2 + x^3
     1 + x + x^2 + x^3 + x^4
     1 + x + x^2 + x^3 + x^4 + x^5
The radius of convergence of the Taylor series is 1, but the function 1/(1-x) is
defined for all x except 1.  This is a good way of graphically demonstrating
that a function may a have a Taylor expansion valid in a smaller domain than the
function is defined.  The interval of convergence is shown on the screen and it
is easy to see that for x < -1 and x > 1, the approximation becomes worse as the
number of terms increases, as distinct from what happens for -1 < x < 1.

F.  arc tan(x),  -1 < x < 1
     This graphs arc tan(x) in the interval -1 < x < 1, and then overlays it
with the polynomials
     x
     x - x^3/3
     x - x^3/3 + x^5/5
     x - x^3/3 + x^5/5 - x^7/7
     x - x^3/3 + x^5/5 - x^7/7 + x^9/9
     x - x^3/3 + x^5/5 - x^7/7 + x^9/9 - x^11/11
The radius of convergence of the Taylor series is 1.

G.  arc tan(x),  -2 < x < 2
     This graphs arc tan(x) in the interval -2 < x < 2, and then overlays it
with the polynomials
     x
     x - x^3/3
     x - x^3/3 + x^5/5
     x - x^3/3 + x^5/5 - x^7/7
     x - x^3/3 + x^5/5 - x^7/7 + x^9/9
     x - x^3/3 + x^5/5 - x^7/7 + x^9/9 - x^11/11
The radius of convergence of the Taylor series is 1, but the function arc tan(x)
is defined for all x.  This is a good way of graphically demonstrating that a
function may a have a Taylor expansion valid in a smaller domain than the
function is defined.  The interval of convergence is shown on the screen and it
is easy to see that for x outside this interval the approximation becomes worse
as the number of terms increases.

H.  sqrt(1 + x),  -1 < x < 1
     This graphs sqrt(1 + x) in the interval -1 < x < 1, and then overlays it
with the polynomials
     1
     1 + x/2
     1 + x/2 - x^2/8
     1 + x/2 - x^2/8 + x^3/16
     1 + x/2 - x^2/8 + x^3/16 - 5x^4/125
     1 + x/2 - x^2/8 + x^3/16 - 5x^4/125 + 7x^5/256
The radius of convergence of the Taylor series is 1.

I.  sqrt(1 + x),  -1 < x < 3
     This graphs sqrt(1 + x) in the interval -1 < x < 3, and then overlays it
with the polynomials
     1
     1 + x/2
     1 + x/2 - x^2/8
     1 + x/2 - x^2/8 + x^3/16
     1 + x/2 - x^2/8 + x^3/16 - 5x^4/125
     1 + x/2 - x^2/8 + x^3/16 - 5x^4/125 + 7x^5/256
The radius of convergence of the Taylor series is 1, but the function sqrt(1+x)
is defined for all x > -1.  This is a good way of graphically demonstrating that
a function may a have a Taylor expansion valid in a smaller domain than the
function is defined.  The interval of convergence is shown on the screen and it
is easy to see that for x outside this interval the approximation becomes worse
as the number of terms increases.

J.  log(1 + x), -2 < x < 2
     This graphs log(1 + x) in the interval -2 < x < 2, and then overlays it
with the polynomials
     x
     x - x^2/2
     x - x^2/2 + x^3/3
     x - x^2/2 + x^3/3 - x^4/4
     x - x^2/2 + x^3/3 - x^4/4 + x^5/5
     x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6
The radius of convergence of the Taylor series is 1, but the function log(1+x)
is defined for all x > -1.  This is a good way of graphically demonstrating that
a function may a have a Taylor expansion valid in a smaller domain than the
function is defined.  The interval of convergence is shown on the screen and it
is easy to see that for x > 1, the approximation becomes worse as the number of
terms increases, as distinct from what happens for -1 < x < 1.

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