Need help for a power series representation question.

Question

f(x)=x/(2x²+1)

I was told to consider 1/(2x²+1) first, and I got ∑(2x²)ⁿ for that... but I'm not sure what to do after that point. If we're writing this as a geometric series, then would x in the numerator just be multiplied with (2x^2)^n?

Answer 1

You're very close on your series. Because the sum of a geometric series is 1/(1-r), your actual series should be ∑(-2x²)ⁿ. Besides that, you should simplify by splitting it into it's distinct parts: ∑(-1)ⁿ2ⁿx²ⁿ. Now bring back the x to make your final answer be ∑(-1)ⁿ2ⁿx²ⁿ⁺¹.