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1. The problem statement, all variables and given/known data

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.

2. Relevant equations

d/dtArctan(t)=1/(1+t^2)

3. 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?

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# Power Series ArcTan?

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