The Closed Form Still Exists!

Calculus Level 2

We know that the geometric progression sum of

1x+x2x3+x4 1 - x + x^2 - x^3 + x^4 - \cdots

can be written as 11+x \dfrac1{1+x} , where 1<x<1-1<x<1.

If we integrate each of these terms with respect to xx, we get the series below. Which of the following is equivalent to the series below?

x12x2+13x314x4+15x5 x -\dfrac12 x^2 + \dfrac13x^3 - \dfrac14 x^4 + \dfrac15x^5 - \cdots

Assume we ignore the arbitrary constant of integration.

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