Taylor Series

Polynomial Approximation

First degree means take 1st derivative, 2 terms total

Taylor

center at some number, c

f(c)+f'(c)(x-c) +...+f^(n)(c)/n!*(x-c)^n

Maclaurin

center at 0

Approximate ln(1.1)

use series ln(1+x)

x=.1

so plugin .1 for the series

Remainder

f(x)=P(x)+R(x)

error = |R(x)|

R(x)=[f^(n+1)(z)/(n+1)!](x-c)^(n+1)

Accuracy

sin(.1)

get real value

get approximation

subtract the 2

Power Series

Radius and Interval of Convergence

IOC

use ratio test

if limit = 0 then converge for all reals

if left with |x-c| set < 1

if limit = infinity converges at center

Remember to not use (-1)^n

test endpoints for ( or [

Radius

(-2,2): R=2

limit is 0, all reals: R=infinity

limit infinity, R=0

Properties

f'(x)

only take derivative on n power with x in term

bring down n, put to n-1

integral

add constant, C

only integrate one with x in term

put to n+1 and divide by n+1

Functions by Power Series

Geometric Power Series

a/(1-r)

goes to: Ear^n

must get it to this form, must have 1 on bottom and minus the x

cannot change the original form

if 1/x, 1/(1-(-x+1)

Operations

if adding 2 together

IOC = intersection of the two

if (-2,2) and (-1,1), IOC=(-1,1)

use partial fractions to get 2 power series

pull out common term

make sure signs are correct, could be + or -

Power series by integration

f(x)=lnx

use f(x)=1/x

integrate the terms of 1/x and you'll get lnx + c

solve for C by letting x=center

or, more easily, integrate the series itself, using rules from 8.8

Taylor and Maclaurin Series

Definition

f^n(c)/n!*(x-c)^n

Convergent series

WRITE OUT

Binomial

f(x)=(1+x)^k

(k(k-1)...(k-n+1)x^n)/n!

Deriving a power series

f(x)=(cosx)^.5

just substitute the x^.5 into the known series

Multiplication and Division

Multiply

multiply the 2 series term by term

gives you the series

Divide

use long division

bottom into top

Approximation from integral

put whatever into the known power series

integrate each of the terms you get from the series

null

do it like you do a definite integral

e^x

1+x+x^2/2!+x^3/3!

x^n/n!

sinx

x-x^3/3!+x^5/5!

(-1)^nx^(2n+1)/(2n+1)!

cosx

1-x^2/2!+x^4/4!

(-1)^nx^(2n)/(2n)!

1/(1-x)

1+x+x^2+x^3

x^n