Question

Use power series operations to find the Taylor series at x = 0 for cos² x. (Hint: cos² x = (1+ cos2x) / 2.)
What is the Taylor series for cos² x at x = 0?

Answer 1

Recall that
\[\cos x=\sum_{k=0}^\infty \frac{(-1)^{k}x^{2k}}{(2k)!}\]
This means
\[\cos 2x=\sum_{k=0}^\infty \frac{(-1)^{k}(2x)^{2k}}{(2k)!}=\sum_{k=0}^\infty \frac{(-4)^{k}x^{2k}}{(2k)!}\]
Add \(1\) and divide by \(2\) and you get the series for \(\cos^2x\).