Quality of Approximation by Geometric Series

x

0.8

polynomial degree

n

0

f(0.800) = 5.0000000

p(0.800) = 1.0000000

error = | f(0.800) - p(0.800) |

= 4.000

0

10

p(x) = 1

This Demonstration shows graphically and numerically the quality of approximating the function values of

f(x)=

1

1-x

for any

x∈(-1,1)

by the Taylor polynomial

p(x)=1+x+

2

x

+...+

n

x

, where

n=0,1,2,…,50

.

Details

Review questions:

Let

x=0.8

and

n=2

. How large is the error?

Let

x=0.8

. For what

n

is the error less than 0.01?

Let

x=-0.4

and

n=50

. How large is the error?

Let

x=0

. Check the error for

n=1,2,3,49,…,50

. (The point

x=0

is the "best" point since it is the expansion point.)

For

n=2

, at which

x

is the error largest?

For

n=50

and

x=0.999

, how large is the error?

For

n=50

and

x=-0.999

, how large is the error?

External Links

Geometric Series (Wolfram MathWorld)

Permanent Citation

Reinhard Simonovits, Bernd Thaller

"Quality of Approximation by Geometric Series"
http://demonstrations.wolfram.com/QualityOfApproximationByGeometricSeries/
Wolfram Demonstrations Project
Published: September 13, 2012