The thing is, not all people is able to see this. Of course, pretty easy when given the partial fraction descomposition and then just do algebra to verify it.
To make what flyingsquirrel did, first note that nothing weird, we just added & subtracted a convenient term. Following up on this, we have that Now, what's the cool thing here: the cool here tell us that in the denominator we have exactly the terms & Hence,
Finally, split this original ratio into two parts, like and the last ratio becomes Then after makin' simple simplifications, we happily get that
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Back to the sum and let So we can see the sum as Add some terms: and so on. Can you see what's goin' on here?