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My homework wants me to find the interval of convergence and the radius of the power series, but it does not give the function, instead it gives the first 4 terms of the series.

The terms are $-x^{9} + \frac{x^{11}}{8} - \frac{x^{13}}{27} + \frac{x^{15}}{64}$

I am confused on how to accomplish this.

Rob Goodall
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1 Answers1

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Well you should see some regularities which enable you to "guess" all the other terms. It is not mathematically strict, but hey, it's your homework:

  1. First, you should see that your series is alternating
  2. Next, notice that the power of $x$ is always increased by 2
  3. Finally notice that denominator is always $\frac{1}{n^3}$

So your series is $$ \sum_{n=1}^{\infty} (-1)^n \frac{x^{7+2n}}{n^3} $$

RedGiant
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