how can i decompose this partial fraction(4x^3)/(x^3-x^2)

This is not a "proper" decomposition problem. Typically the degree of the denominator is greater than the degree of the numerator...

but...we can still simplify...
notice that this expression can be factored:

from here we notice that by polynomial long division (don't know how to show using this editor) that:

so our expression simplifies to...



:)

Partial Fraction Decomposition

Partial Fraction Decomposition

Series Convergence

This example is nice because we usually cannot get to the sums of these series so easily. But after decomposing the operand, we expand it and convince ourselves that every term cancels with a successor except for the first term.

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