Power Series ArcTan: Analyzing the Expansion & Convergence

[sammiekurr]

Power Series ArcTan?

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?

 

[HallsofIvy]

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

No. You have the formula right but when n= 0, 4(-1)^n t^2n is 4. The first four terms are 4- 4t^2+ 4t^4- 4t^6.

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?

Why is it "not in the standard form of a power series"?

You have that 4/(1+ t^2)= 4\sum_{n=0}^\infty (-1)^n t^{2n} and can integrate "term by term".