Convergent/Divergent Chart
Using the "Nth Term Test for Divergence," if the limit as n approached infinity of An does not equal zero what happens?
The series diverges.
What are sequences?
A collection of numbers with a one-to-one correspondence with ∞ positive integers.
What is it called when a function is differentiable at a point c then it can also be approximated near c by its tangent line?
The linear approximation to f at the point c.
What is a "Lagrange Error Bound?"
It is the worst-case scenario for how far a Taylor approximation is from the actual function at a point.
What is a "Maclaurin Polynomial"?
It is a Taylor polynomial centered about x=0.
What does it mean when a series converges conditionally?
The absolute value of an diverges, but an converges.
What is an Infinite Series?
The sum of all the terms as n approaches infinity.
What could you name a first-degree polynomial of x?
P1(x)
Use a third-degree Taylor polynomial on the interval [0,1] for ex centered at x=0 to approximate e1. What is the error bound of this approximation?
R3(x) < .1132617
Find the third-degree Maclaurin polynomial for F(x)=e2x. Evaluate at f(0.2) and p3(0.2).
f(0.2)=1.49182
f3(0.2)=1.490666
Using the "p-series" test, if an=1/np, n>1 and lrl<1 does the series converge or diverge?
The series converges.
What is a Partial Sum?
The sum of the first 'n' terms.
If f(x) is a differentiable function, then an approximation of f centered about x=c can be modeled by?
pn(x)=f(c)+f'(c)(x-c)+f''(c)(x-c)2/2!+f'''(c)(x-c)3/3!+ ... +fn(c)(x-c)n/n!
What is the smallest order Taylor Polynomial centered at x=1 which will approximate ex-1 on the interval [0,1] with a Lagrange error bound less than 1?
n=5
The fourth degree Maclaurin polynomial for cos x is given by p4(x)=1-(x2/2!)+(x4/4!). If this polynomial is used to approximate cos(0.2), what is the Lagrange error bound?
R4(x) < 2.667 x 10-6
Does {an}={4n/5n} converge or diverge?
Converges because {an} gets closer and closer to zero.
What is the fourth term of the following sequence: an=1/2n ?
1/16
Find the fourth-degree Taylor polynomial for f(x)=ln x centered at x=1.
P4(x) = (x-1)-.5(x-1)2+(x-1)3/3-.25(x-1)4
If the Taylor Polynomial for approximating cos x is given by 1-(x2/2!)+(x4/4!), what is the upper bound for the error in the approximation of cos(0.3)?
2.025 x 10-5
If c=0 it is a...
Maclaurin Series
What qualifications does a series need to converge while using the "Integral Test?"
an=f(n) will be continuous, it will be positive, and it will be decreasing on [a,∞]. bn must also converge.
What is the explicit formula for the nth term of the following sequence: {an}={3,6,12,24,48,...}?
an=3(2)n-1
Let f be a function with third derivative f'''(x)=(8x+2)(3/2). What is the coefficient of (x-2)4 in the fourth-degree Taylor polynomial for f about x=2?
f(4)(2)=36√2
The function f has derivatives of all orders for all real numbers, f(4)(x)=ecosx. If the third-degree Taylor Polynomial for f about x=0 is used to approximate f on the interval [0,1], what is the Lagrange error bound?
.11236
Find the Maclaurin series for sin x.
sin x= x-(x3/3!)+(x5/5!)-(x7/7!)+ ... ((-1)nx2n+1)/(2n+1)!