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I am having some trouble finding the first three nonzero terms for the maclaurin series of $\frac{\cos2x-1}{x^2}$ using the maclaurin series for $\cos2x$.

So far I have the maclaurin series for $\cos2x$ being:$$\cos2x=1-\frac{4x^2}{2!} + \frac{16x^4}{4!} - \frac{64x^6}{6!}\dots$$However I am not sure how to proceed in finding the macluarin series for $\frac{\cos2x-1}{x^2}$ now.

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2 Answers

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Maclaurin series can be treated as infinite polynomials (most of the time), so we have$$\cos2x-1=- \frac{4x^2}{2!} + \frac{16x^4}{4!} - \frac{64x^6}{6!}+\dots$$$$\frac{\cos2x-1}{x^2}=- \frac{4}{2!} + \frac{16x^2}{4!} - \frac{64x^4}{6!}+\dots$$$$=-2+\frac23x^2-\frac4{45}x^4+\dots$$

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Since \begin{eqnarray} cos(2x) = 1 - \frac{4x^2}{2!} + \frac{16x^4}{4!} - \frac{64x^6}{6!} ... \end{eqnarray}$$\cos (2x) - 1 = \frac{4x^2}{2!} + \frac{16x^4}{4!} - \frac{64x^6}{6!} ...$$$$\frac{\cos(2x)}{x^2} = \frac{4}{2!} + \frac{16x^2}{4!} - \frac{64x^4}{6!} ...$$$$=2 + \frac{16x^2}{4!} - \frac{64x^4}{6!} ...$$

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