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The following is a taylor series of a function centered at 8.

$$\sum_{n=1}^{\infty}\frac{(-1)^n(x-8)^n}{7^n(n+8)}$$

I am trying to ind the radius of convergence of R of this series.

My guess is that the radius of convergence is $\infty$ because we have $(\frac{-1}{7})^n$, which approaches to 0 as n approaches $\infty$. Since we have a case of convergence, I suppose that the whole function should be convergent for whatever value of x. This is my attempt at answering the question, however, the answer seems to be incorrect.

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  • $\begingroup$ Are you familiar with the ratio test ? (It can be done using it). $\endgroup$
    – mich95
    Apr 7, 2015 at 2:39
  • $\begingroup$ You can't simply guess the radius of convergence.. If you are doing this in a class you must be familiar with the ratio test as it its one of the only ways to do this question.. $\endgroup$
    – Quality
    Apr 7, 2015 at 2:40
  • $\begingroup$ Ah yes, I've solved it using the ratio test, Thank You. The summation variable k was a typo, sorry! $\endgroup$
    – TheValars
    Apr 7, 2015 at 2:41

1 Answer 1

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Let $x_{n}$ be the sequence inside the sum. $\vert \frac{x_{n+1}}{x_{n}}|=\vert \frac{(x-8)(n+8)}{7(n+9)} \vert$. Hence, as n goes to infinity $\vert \frac{x_{n+1}}{x_{n}}|$ goes to $\vert \frac{(x-8)}{7} \vert$. The radius of convergence is 7.

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