Power series question 2

[miniskus]

The other power series question wasn't hard once I got help splitting it up, but this one I'm not sure what to do, due to a polynomial in the numerator. I need to find the radius of this power series.

1/(1-x) + (2x+3)/(1+x^2).

The first term is easy for me to solve but it is the second that is giving me trouble at the end. I recognize that the second term is equal to (2x+3) times the series -x^2n, meaning that abs ((2x+3)x^2)<1, but i don't know if this is right! If this paragraph doenst make sense, the original equation is right.

 

[soroban]

Hello, miniskus!

I don't know if my approach is acceptable . . .

This one, I'm not sure what to do, due to a polynomial in the numerator.
I need to find the radius of this power series.

1/(1-x) + (2x+3)/(1+x²)

The first term is easy for me to solve but it is the second that is giving me trouble at the end.

The first term is: . 1/(1-x) .= .1 + x + x² + x³ + . . .

I used long division on the second term and got this patterned series:
. . (2x + 3)/(1 + x²) .= .3 + 2x - 3x² - 2x³ + 3x⁴ + 2x⁵ - 3x⁶ - 2x⁷ + ...


Adding the two series, I got:
. . 4 + 3x - 2x² - x³ + 4x⁴ + 3x⁵ - 2x⁶ - x⁷ + 4x⁸ + 3x⁹ - 2x¹⁰ - x¹¹ + ...

= .(4 + 3x - 2x² - x3) + x⁴(4 + 3x - 2x² - x³) + x⁸(4 + 3x - 2x² - x³) + ...


This is a gemetric series with first term, a = (4 + 3x - 2x² - x³) and common ratio, r = x⁴.

The series converges if |r| < 1 . . . that is: .x⁴ < 1 . . ---> . . |x| < 1

. . and there is our radius of convergence.