Taylor Series
8.7
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
8.8
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
8.9
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
8.10
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
Common Series
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