This is from the teacher's notes(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

[tex]f(x) = \frac{x}{{9 + x^2 }} = \frac{x}{9} \cdot \frac{1}{{1 - \left( { - \frac{{x^2 }}{9}} \right)}} = \frac{x}{9}\sum\limits_{n = 1}^\infty {\left( { - \frac{{x^2 }}{9}} \right)^n } = \sum\limits_{n = 1}^\infty {\frac{{( - 1)^n x^{2n + 1} }}{{9^{n + 1} }}} [/tex]

I can see distributing the n inside the parenthesis, to -1, x^2, and 9. But what's the justification for chaning it to n+1 for x^2 and for 9?

The next step is

This converges for [tex]\left| { - \frac{{x^2 }}{9}} \right| < 1\,\,\,\,\,\,or\,\,\,\,x^2 < 9\,\,\,\,\,\,\,\,\,\,\, - 3 < x < 3[/tex]

(−3, 3)

So what was the point in doing the last step in my first tex, if she just resorted to the 2nd to last step to determine the interval of convergence?

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Power series

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