Thread: Evaluating an indefinite integral as an infinite series

Greetings! I'm not sure how to approach this question, it's asking that I rewrite this:

∫(sin(x)-x)/x^3 dx

as an infinite series. Any help is appreciated!

Write sin(x) as a Taylor's series. Then you can subtract x, divide by and integrate term by term.

Ok let me see...
sinx as a Taylor series would be the sum of ((-1)^n/(2n+1)!)*x^(2n+1)

so then would we have (((-1)^n/(2n+1)!)*x^(2n+1)/x^3) - (x/x^3)?

Yes, and surely you can now simplify...

It might make more sense if you write it out as .
Then
and
Integrate that.

Ah! Alright that makes much more sense now. Thank you very much!