Power series question 2

miniskus

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Jun 13, 2005
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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.
 
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+x<sup>2</sup>)

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 + x<sup>2</sup> + x<sup>3</sup> + . . .

I used long division on the second term and got this patterned series:
. . (2x + 3)/(1 + x<sup>2</sup>) .= .3 + 2x - 3x<sup>2</sup> - 2x<sup>3</sup> + 3x<sup>4</sup> + 2x<sup>5</sup> - 3x<sup>6</sup> - 2x<sup>7</sup> + ...


Adding the two series, I got:
. . 4 + 3x - 2x<sup>2</sup> - x<sup>3</sup> + 4x<sup>4</sup> + 3x<sup>5</sup> - 2x<sup>6</sup> - x<sup>7</sup> + 4x<sup>8</sup> + 3x<sup>9</sup> - 2x<sup>10</sup> - x<sup>11</sup> + ...

= .(4 + 3x - 2x<sup>2</sup> - x3) + x<sup>4</sup>(4 + 3x - 2x<sup>2</sup> - x<sup>3</sup>) + x<sup>8</sup>(4 + 3x - 2x<sup>2</sup> - x<sup>3</sup>) + ...


This is a gemetric series with first term, a = (4 + 3x - 2x<sup>2</sup> - x<sup>3</sup>) and common ratio, r = x<sup>4</sup>.

The series converges if |r| < 1 . . . that is: .x<sup>4</sup> < 1 . . ---> . . |x| < 1

. . and there is our radius of convergence.
 
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