Calculus - Taylor and Maclaurin Series

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Taylor Series

Common Series

1/(1-x)

1+x+x^2+x^3

x^n

cosx

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

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

sinx

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

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

e^x

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

x^n/n!

8.10

Taylor and Maclaurin Series

Approximation from integral

put whatever into the known power series

null

do it like you do a definite integral

integrate each of the terms you get from the series

Multiplication and Division

Divide

use long division

bottom into top

Multiply

multiply the 2 series term by term

gives you the series

Deriving a power series

f(x)=(cosx)^.5

just substitute the x^.5 into the known series

Binomial

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

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

Convergent series

WRITE OUT

Definition

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

8.9

Functions by Power Series

Power series by integration

f(x)=lnx

use f(x)=1/x

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

solve for C by letting x=center

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

Operations

use partial fractions to get 2 power series

make sure signs are correct, could be + or -

pull out common term

if adding 2 together

IOC = intersection of the two

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

Geometric Power Series

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

cannot change the original form

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

a/(1-r)

goes to: Ear^n

8.8

Power Series

Properties

integral

put to n+1 and divide by n+1

only integrate one with x in term

add constant, C

f'(x)

bring down n, put to n-1

only take derivative on n power with x in term

Radius and Interval of Convergence

Radius

limit infinity, R=0

limit is 0, all reals: R=infinity

(-2,2): R=2

IOC

test endpoints for ( or [

use ratio test

Remember to not use (-1)^n

if limit = infinity converges at center

if left with |x-c| set < 1

if limit = 0 then converge for all reals

8.7

Polynomial Approximation

Accuracy

sin(.1)

subtract the 2

get approximation

get real value

Remainder

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

error = |R(x)|

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

Approximate ln(1.1)

use series ln(1+x)

so plugin .1 for the series

x=.1

Maclaurin

center at 0

Taylor

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

center at some number, c

First degree means take 1st derivative, 2 terms total