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## Homework Statement

F(x)=∫(0 to x) tan^(-1)t dt. f(x)= infinite series ∑n=1 (-1)^(en)(an)x^(pn)?

en=?

an=?

pn=?

I know en = n-1

## Homework Equations

## The Attempt at a Solution

Start with the geometric series

1/(1 - t) = ∑(n=0 to ∞) t^n.

Let t = -x^2:

1/(1 + x^2) = ∑(n=0 to ∞) (-1)^n * x^(2n).

Integrate both sides from 0 to x:

arctan x = ∑(n=0 to ∞) (-1)^n * x^(2n+1)/(2n+1).

Now that we have a series for arctan x...

f(x) = ∫(0 to x) arctan t dt

= ∫(0 to x) [∑(n=0 to ∞) (-1)^n * t^(2n+1)/(2n+1)] dt

= ∑(n=0 to ∞) (-1)^n * [∫(0 to x) t^(2n+1)/(2n+1) dt]

= ∑(n=0 to ∞) (-1)^n * x^(2n+2)/[(2n+1)(2n+2)].

Shifting indices up one unit, we have

f(x) = ∑(n=1 to ∞) {(-1)^(n-1)/[(2n-1)(2n)]} x^(2n).

This gives me

en = n-1

an=1/(2n(2n-1))

pn = 2n

however an and pn are wrong. where am i going wrong?