I am currently working on the power series for a homework assignment. I have to find the radius of convergence for the function $$\frac{10}{1+64x^2}$$

By setting up the


part of the expression as a derivative of


I was able to expand the series term by term and get the correct co-efficients.

The series is:

$$\frac{10}{1+64x^2} = 10 - 640x^2 + 40960x^4 - 2621440x^6 ...$$

This series is correct, since my homework portal accepts it. In order to find the radius of convergence, I need to be able to put this series in the form of a summation notation so that I can apply the ratio tests. How do I extract this series' summation notation? How would I account for the 0 co-efficients for the terms with degree $1, 3$, and so forth (odd-terms). I know it involves an alternating series somewhere, but I am not sure how to account for the terms whose co-efficients are zero, when it comes to getting the summation form notation.

  • $\begingroup$ Hint: Is there something that you can multiply each term by to get the next? $\endgroup$ – Archaick Apr 12 '15 at 6:21
  • $\begingroup$ Please use latex for maths text.. See here. $\endgroup$ – mattos Apr 12 '15 at 6:23
  • $\begingroup$ 8x? I will learn how to use LaTeX, I'm new to this website and haven't learned it yet. Thanks for the resource. $\endgroup$ – Ferreroire Apr 12 '15 at 6:26

$$\begin{align} \frac{10}{1+64x^2} &= 10 - 640x^2 + 40960x^4 - 2621440x^6 ... \\ &= 10 [1 - 64x^{2} + 4096x^{4} - 262144x^{6} ... ]\\ &= 10 [(-1)^{0}(8x)^{0} + (-1)^{1}(8x)^{2} + (-1)^{2}(8x)^{4} + ...] \\ &= 10 \sum_{k = 0}^{\infty} (-1)^{k} (8x)^{2k} \\ \end{align}$$

  • $\begingroup$ Awesome, I was blind! The power ^2k explains so much, thanks! I thought along the lines of how to get it to cycle through the odd terms as well. $\endgroup$ – Ferreroire Apr 12 '15 at 6:33

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