- #1
sammiekurr
- 2
- 0
Power Series ArcTan?
Let f be the function given by f(t) = 4/(1+t^2) and G be the function given by G(x)= Integral from 0 to x of f(t)dt.
A) Find the first four nonzero terms and the general term for the power series expansion of f(t) about x=0.
B) Find the first four nonzero terms and the general term for the power series expansion of G(t) about x=0.
C) Find the interval of convergence of the power series in part (B). Show the analysis that leads to your conclusion.
d/dtArctan(t)=1/(1+t^2)
A) a=4, R=-t^2. f(t)=Sum from n=1 to infinity of 4 * (-1)^n * t^2n
First four terms: -4t^2 + 4t^4 - 4t^6 + 4t^8
B) Integral from 0 to x of 4/(1+t^2)dt = 4arctan(t) from 0 to x = 4arctan(x)
Now I don't know where to go from here. I don't know how to write the power series for the antiderivative of the original power series, since it is not in the standard form of a power series. Can anybody help?
Homework Statement
Let f be the function given by f(t) = 4/(1+t^2) and G be the function given by G(x)= Integral from 0 to x of f(t)dt.
A) Find the first four nonzero terms and the general term for the power series expansion of f(t) about x=0.
B) Find the first four nonzero terms and the general term for the power series expansion of G(t) about x=0.
C) Find the interval of convergence of the power series in part (B). Show the analysis that leads to your conclusion.
Homework Equations
d/dtArctan(t)=1/(1+t^2)
The Attempt at a Solution
A) a=4, R=-t^2. f(t)=Sum from n=1 to infinity of 4 * (-1)^n * t^2n
First four terms: -4t^2 + 4t^4 - 4t^6 + 4t^8
B) Integral from 0 to x of 4/(1+t^2)dt = 4arctan(t) from 0 to x = 4arctan(x)
Now I don't know where to go from here. I don't know how to write the power series for the antiderivative of the original power series, since it is not in the standard form of a power series. Can anybody help?