Lagrange Error Bound and Accuracy
Use Taylor's Inequality the error bounds of the approximation
eapprox 1+1+1/(2!)+1/(3!)+1/(4!)
approx0.009948
Find the radius and interval of convergence for the series
sum_(n=1)^oo (-1)^n nx^n
Radius: 1
Interval: -1<x<1
Write the basic Maclaurin series
T(x) centered at x=c
sum_(n=0)^oo (f^nc)/(n!)(x-c)^n
Find the Maclaurin series representation for the series and the radius of convergence
f(x)=1/(1-x^3)
sum_(n=0)^oo x^(3n)
Radius: 1
Use Taylor's Inequality the error bounds of the approximation
cos(0.4) approx 1-(0.4)^2/(2!)+(0.4)^4/(4!)
approx 5.01*10^-6
Find the radius and interval of convergence for the series
sum_(n=1)^oo n!(x+2)^n
Radius: 0
Interval: -2
Write the basic Maclaurin series in general form
f(x)=1/(1+x)
sum_(n=0)^oo (-1)^nx^n
for |x|<1
Find the Maclaurin series representation for the series and the radius of convergence
f(x)=x^2e^(-x)
sum_(n=0)^oo ((-1)^nx^n)/(n!)
Radius:
oo
Find a 4th degree Taylor polynomial for f(x) about x=4, then find the Lagrange error bound on the interval [4, 4.5]
4th degree Taylor polynomial:
ln4+1/4(x-4)-1/32(x-4)^2+1/192(x-4)^3-1/1024(x-4)^4
Lagrange error bound:
greater than or equal to
6.1035*10^-6
Find the radius and interval of convergence for the series
sum_(n=1)^oo ((2x-3)^n)/(n5^n
Radius:
5/2
Interval: -1 less than or equal to x<4
Find the Maclaurin series for
f(x)=3e^(-2x)
Write the first three terms and the general term
3-6x-(12x^2)/(2!)+...+((-1)^n3(2x)^n)/(n!
Find a Taylor series about x=0 for the series
int(sinx)/xdx
C+sum_(n=0)^oo ((-1)^nx^(2n+1))/((2n+1)(2n+1)!
Given
f(x)=sqrt4
Write a 2nd degree Taylor polynomial for f(x) about x=4, then find the Lagrange error bound for the approximation on the interval [4, 4.1]
2nd degree Taylor polynomial:
2+1/4(x-4)-1/64(x-4)^2
Lagrange error bound approximation:
greater than or equal to
1.953*10^-6
Find the radius and interval of convergence for the series
sum_(n=0)^oo x^n/(2n+1)
Radius: 1
Interval: -1 less than or equal to x<1
Use a known Maclaurin series to evaluate
lim_(x->0) (cosx-1+x^2/2)/x^4
1/24
Find a Taylor series about x=0 for the series
int(e^(x^2))/xdx
ln|x|+sum_(n=1)^oo (x^(2n))/(2n*n!)+ C
Let f(x) be a function that is continuous and differentiable at all real numbers, and f(2)=1, f'(2)=5, f"(2)=7, and f'"(2)=-3.
Write a 3rd order Taylor polynomial for f(x) about x=2, also find the Lagrange error bound for the approximation if
f^(4)(x)
greater than or equal 4
3rd order Taylor polynomial:
1+5(x-2)+7/2(x-2)^2-1/2(x-2)^3
Lagrange error bound:
greater than or equal to
1.666*10^-5
Find the radius and interval of convergence for the series
sum_(n=1)^oo ((-1)^nx^n)/(n^2+2
Radius: 1
Interval -1 less than or equal to x less than or equal to 1
Express
intsinx^2
as a power series
C+sum_(n=0)^oo ((-1)^nx^(4n+3))/((4n+3)(2n+1)!
Find a power series for
f(x)=ln(2+x)
f(x)=ln2+sum_(n=0)^oo ((-1)^n)/(n+1)(x/2)^(n+1)