Calc Week 6

Question 1

power series

Answer 1

x is a variable
a power series is an infinite series of the form

sigma from n =0 to inf
C sub n X^n

where {C sub n} is a sequence of coefficients

Question 2

shift operations

Answer 2

adding or subtracting from the argument x in f(x)

Question 3

power series centered at a

Answer 3

a is a constant
a power series in x centered at a is an infinite series of the form

sigma from n=0 to inf
C sub n (x-a)^n

Question 4

interval of convergence

Answer 4

set of all values of x such that the series converges

Question 5

radius of convergence

Answer 5

if series converges for all x, then we say the interval of convergence is infinite, R = inf

if series converges at a single point, we say that R=0

Question 6

power series convergence trichotomy

Answer 6

consider a power series centered at a

sigma from n=0 to inf
C sub n(x-a)^n

there are 3 possibilities

1. the series converges for all x
2. the series converges for only a single value of x, which is x =a
3. the series converges on a finite interval
   there is a R>0 such that
   -the series converges if |x-a| < R
   -the series diverges if |x-a| > R

Question 7

steps for finding radius of convergence and interval of convergence

Answer 7

1. apply ratio test with absolute values
2. set L < 1 and solve for x (the L you found from 1) for radius of convergence

3.check if the series converges at the endpoints
-plug in x and see what the series does
- to include or not to include in the interval of convergence

Question 8

differentiating power series

Answer 8

differentiate each term and the sum
does not change the radius of convergence
could affect the endpoints

Question 9

integrating power series

Answer 9

integrate each term and the sum
does not change the radius of convergence but could affect the endpoints

Question 10

geometric series and the sum of the alternating harmonic

Answer 10

work on one note

??by integrating a power series, you get the sum

Question 11

Leibniz formula and the power series of arctan

Answer 11

work on one note

1 - 1/3 + 1/5 -1/7 + 1/9 - … = pi/4

Question 12

nth formula taylor polynomial

Answer 12

let f(x) be a function whose 1st n derivatives exist at the point x = a. The nth degree taylor polynomial of f at the point x = a is

P sub n (x) = sigma from k=0 to n
(f ^(k) (a) / k! ) (x-a)^k

expand

P sub n (x) =
f(a) + f’(a)(x-a) + f’‘(a)/2! (x-a)^2 + f’’‘(a)/3! (x-a)^3 +…+
f^n(a)/n! (x-a)^n

Question 13

maclaurin polynomial of f (of degree n)

Answer 13

if a = 0 in the nth degree taylor polynomial

Question 14

find the P sub n (x)

Answer 14

write the polynomial equation up to n

find table of derivatives

plug in x into those derivatives

plug in the values in P sub n(x)

Question 15

meaning of the value of P sub n(x)

Answer 15

approximation of f(x) provided that x is near a