chapter 9 (Power series)

Question 1

general form of a power series

Answer 1

sigma(k to inf) ck(x-a)^k

Question 2

ck in power series

Answer 2

the coeffiecients

Question 3

a in a power series

Answer 3

the offset of the center

Question 4

power series

Answer 4

an infinite series whoos terms include powers of a variable

Question 5

linear approximation of f(x) at a

Answer 5

L(x) = f(a) +f’(a)(x-a)

Question 6

quadratic approximation of f(x) at a

Answer 6

Q(x) = f(a) +f’(a)(x-a) + f’‘(a)/2(x-a)^2

Question 7

the nth order taylor polynomial

Answer 7

sigma(k to n) ck(x-a)^k

Question 8

ck in a Taylor polynomial

Answer 8

kth derivative of f(a)/k!

Question 9

remainder for an nth order polynomial with M

Answer 9

Rn <= M((|x-a|^n+1)

Question 10

remainder formula

Answer 10

f^n+1(c)/(n+1)! (x-a)^n+1

Question 11

differences between geometric series as a series and as a power series

Answer 11

basically the r in a/1-r becomes an x

so the power series is 1/1-x

Question 12

power geometric series

Answer 12

1/1-x = sigma x^k

Question 13

find a power series representation for x^4/1-x using a geometric power series

Answer 13

sigma(k to inf) (x^4)(x^k)

Question 14

deriving or intergrating geometric power series to find power series to match ones

Answer 14

basically take the derivative or integral of a power series if the derivative or integral gets you closer to what you need to approximate it to

Question 15

when you derive or integrate the non sigma side, what must be done to the sigma side

Answer 15

the same exact process

Question 16

interval of convergence

Answer 16

this is the interval that the power series is still converghent uppon

Question 17

how to find the interval of convergent

Answer 17

use the root or ratio test.

Question 18

root test

Answer 18

p= lim of kthsqrt(ak)

Question 19

ratio test

Answer 19

l= lim of a(k+1)/ak

Question 20

if the root or ratio limit is 0

Answer 20

the radius is inf

the interval is also inf

Question 21

if the root or ratio limit is inf

Answer 21

the radius is 0

the interval does not exist. diverges everywhere

Question 22

if the root or ratio limit is anything else

Answer 22

put it in absolute values and solve for x. then find the radius and the 2 potential endpoints

Question 23

what to do with two endpoints for the interval of convergence

Answer 23

test each with different series tests. if they converge then the endpoint is inclusive. if they diverge the endpoint is noninclusive

Question 24

makalaureans series

Answer 24

a taylor series that has an a of 0

Question 25

finding the taylor series for a function

Answer 25

make the collums

1. the kth derivative
2. the kth derivative evaluated at a
3. f^k(a)/K!
4. that multiplied by (x-a)^k

find the damn pattern

Question 26

how to get a taylor series to alternate values and skip odd values used for cosine

Answer 26

x must be raised to the 2k
it must be 2k!
and k must start at 0

Question 27

how to get a taylor series to alternate values and skip even values
used for sin

Answer 27

x must be raised to the 2k+1
it must be (2k+1)!
and k must start at 0

Question 28

taylor series for 1/1-x

Answer 28

sigma (k=0 to inf) x^k

Question 29

taylor series for e^x

Answer 29

sigma (k=0 to inf) x^k/k!

Question 30

taylor series for ln(1+x)

Answer 30

sigma (k=1 to inf) ((-1^k+1)x^k/k for -1

Question 31

taylor series for sinx

Answer 31

sigma (k=0 to inf) ((-1^k)(x^2k+1)/(2k+1)!

Question 32

taylor series for cosx

Answer 32

sigma (k=0 to inf) ((-1^k)(x^2k)/(2k)!

Question 33

approximate

Answer 33

use the nth order taylor polynomial.
plug in a value close to what you are approxximating
calc

Question 34

estimate the remainder

Answer 34

find what f^n+1 is bound by

find M then plug in with the given n to the < one