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I am required to find the radius of convergence for the function $$f(x) = 5x^3 - 6x^2 - 7x + 6$$ by first finding its Maclaurin series.

I found the Maclaurin series to be $$ 6 - 7x -6x^2 + 5x^3...$$

I am having trouble figuring out what the summation form notation of this series should look like. I would need this to apply the ratio test and find the radius of convergence.

Some guidelines on figuring out the summation notation for any such series would also be appreciated. Thanks!

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  • $\begingroup$ What are the values of the coefficients for the terms $x^4$, $x^5$... $\endgroup$ Apr 14 '15 at 20:33
  • $\begingroup$ There shouldn't be any $\ldots$ on your Maclaurin series, for the Maclaurin series of any single-variable polynomial can only be the polynomial itself. As such, the series isn't divergent for any real $x$ (i.e., the series is convergent everwhere over $x \in \mathbb{R})$. $\endgroup$
    – Brian Tung
    Apr 14 '15 at 20:38
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The coefficients on the $x^4$ term and beyond are all zero so the ratio test isn't suitable here. On the other hand you can use the root test to show the radius of convergence is infinite. That is, the finite series converges to $f$ everywhere on the line.

Looking back, the root test isn't really necessary. A finite power series is clearly convergent at every point. To what? Itself.

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  • $\begingroup$ Wouldn't we need the summation notation to apply the root test? Also, could you elaborate on what you meant by the last statement, I didn't quite understand. $\endgroup$
    – Ferreroire
    Apr 14 '15 at 20:36
  • $\begingroup$ No, all you need is a formula for the coefficients, which you have: $a_k = 0$ if $k \ge 4$. The definition of a convergent series is all you really need to know that the maclaurin series of a polynomial is just that polynomial, convergent on the whole line. $\endgroup$
    – Umberto P.
    Apr 14 '15 at 20:38

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