Power Series
What are the two things to find when doing Power Series?
Interval of convergence & radius of convergence
Find the IoC:
g(x)=(x3)/(1+x7)
IoC: (-1,1)
What makes a Taylor series into a Maclaurin series?
When a=0
Consider the following function.
f(x)=1/x, a=1, n=2, 0.7≤x≤1.3
Approximate f by a Taylor polynomial with degree n at the number a.
f(x)=1-(x-1)+(x-1)2
What planet has the strongest gravity?
Jupiter
What is the primary test to find the radius of convergence?
Ratio Test
Find a power series representation:
g(x)= (x3)/(1+x7)
∑ (-1)nx7n+3
n=0 to ∞
Find the function represented by the given power series:
∑ (-1)n(x8n)/(n!)
ex=∑ (xn)/n! for all x
When Amazon started what was the only product they sold?
Books
Find the radius & interval of convergence
∑ (xn/n4n)
n=1 to ∞
RoC: 4
IoC: [-4,4)
Find the power series representation:
f(x)= 6/(1+7x4)
∑ 6(-7)nx4n
n=0 to ∞
Find the Taylor series for f centered at 5 if:
f(n)(5)=((-1)nn!)/(4n(n+3))
∑ ((-1)n(x-5)n)/(4n(n+3))
A car is moving with speed 50 m/s and acceleration 4 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second.
52 m
What singer holds the most Grammy nominations?
Beyonce
Find the RoC & IoC.
∑ ((n+1)(x-2)n)/(2n+1)!
n=0 to ∞
RoC: ∞
IoC: (-∞ ,∞ )
Find the IoC:
f(x)= (3x2)/(5-2∛x)
-125/8 < x < 125/8
What is the radius of convergence R of the Taylor series?
f(n)(5)=((-1)nn!)/(4n(n+3))
R=4
How long is a marathon?
26.2 miles
Find the RoC & IoC:
∑ ((x-8)n)/(n4+1)
n=0 to ∞
RoC: 1
IoC: [7,9]
Find the power series representation:
f(x)= (3x2)/(5-2∛x)
∑ (3/5)(2/5)nx1/3n+2
Find the Maclaurin series:
f(x)= 9(1-x)-2
9∑(n+1)xn
Consider the following function
f(x)=sin(x), a=pi/6, n=4, 0≤x≤pi/3
Approximate f by a Taylor polynomial with degree n at the number a.
1/2 + √3/2(x-pi/6) - 1/4(x-pi/6)2 - √3/12(x-pi/6)3 + 1/48(x-pi/6)4
What is the closest living relative to the T-Rex?
Chicken