Given that the remainder term for the nth - order Taylor polynomial for f centered at a is Rn(x) = f^(n+1)(c) (x-a)^(n+1) / (n+1)! , then the remainder term for the 3rd -order Taylor Polynomial for sin(2x) centered at a=0 satisfies |R3(x)| is less than or equal to ___?
What is 2x^4/3?
100
Find the coefficient of (x- 1/2)^3 in the Taylor series for f(x) = ln(2x) centered at x = 1/2.
What is 8/3?
100
The Maclaurin series for f(x)=sinx is given by for the sum fom 0 to infinity of (-1)^k x^(2k+1) / (2k +1)! What is the Maclaurin series for g(x) = x^3sinx?
What is the sum from 0 to infinity of (-1)^k x^(2k+4) / (2k+1)!
100
The power series of the sum from 0 to infinity of (-1)^(k+1) (x-5)^k / k5^k is centered at a = 5 with a radius of convergence R=5. What is the interval of convergence?
What is (0,10]
100
Is the sum from 1 to infinity of (-1)^k e^k / pi^k absolutely convergent, conditionally convergent or divergent?
What is converges absolutely.
200
use a the first 4 terms of a taylor polynomial to approximate the value of ln(1.2)
What is 0.1822666
200
Find the coefficient of (x - pi/4)^3 in the Taylor series for f(x) = cosx centered at pi/4?
What is 2^(1/2) / 12?
200
Recall that the Maclaurin series for f(x) = tan^(-1)(x) is given by the sum from 0 to infinity of (-1)^k x^(2k+1) / (2k + 1) for -1 less than or equal to x less than or equal to 1. What is the sum from 0 to infinity of (-1)^k / (3^(1/2))^(2k+1) (2k + 1) ?
What is pi/6
200
Analytically determine the radius of convergence, R, for the power series the sum from 0 to infinity of x^(k+1) / (2k+1)!
What is infinity
200
Use the integral test to determine if the series the sum from 1 to infinity of k / (k^4 + 1) converges or diverges.
What is converges.
300
estimate (1.1)^(1/2) using a taylor polynomial of f(x)=(x+1)^(1/2) centered at 0.
What is 839/800.
300
Find the nth - order Taylor polynomials for n = 0, 1, 2, 3 for f(x) = 1/(x+1) centered at a = 1.
The Maclaurin series for f(x) = 1/ (1+x) is given by the sum from 0 to infinity (-1)^k x^k for -1
What is the sum from 0 to infinity of (-1)^k kx^(k-1)
300
Use a test from the attached Series Tests Summary sheet to analytically determine the radius of convergence, R, for the power series the sum from 1 to infinity of (-1)^(k+1) k(x-3)^k / 2^k
What is R = 2.
300
Use a test from the attached Series Tests Summary sheet to analytically determine if the series of the sum from 1 to infinity of (-1)^(k+1) (k^4 +1) / (k^6 +3k^2 +k) converges absolutely, conditionally or diverges.
What is absolutely converges by limit comparison of the sum of |An|.