The sum of the series $\cos x-\frac{1}{2} \cos ^2 x+\frac{1}{3} \cos ^3 x-\frac{1}{4} \cos ^4 x+\ldo

Question

The sum of the series $\cos x-\frac{1}{2} \cos ^2 x+\frac{1}{3} \cos ^3 x-\frac{1}{4} \cos ^4 x+\ldots$ is

Professor PC Thomas Classes · Answer

Maclaurin series for $\ln(1+t) = t - t^2/2 + \dots$ Sum $= \ln(1+\cos x) = \ln(2\cos^2(x/2)) = \ln 2 + 2\ln|\cos(x/2)|$.