Find Taylor series for $\cosh z \cos z$.
$\cos z = \cosh iz$ and $\cosh z \cosh iz = \dfrac{1}{2}(\cosh (i+1)z + \cosh (i-1)z)$ and finally $$\cosh z \cos z = \dfrac{1}{2}\sum_{n=0}^\infty {\dfrac{(i+1)^{2n} + (i-1)^{2n}}{(2n)!}z^{2n}} = \sum_{n=0}^\infty {\dfrac{i^n 2^n}{(2n)!}z^{2n}}$$ but surprisingly the answer in the textbook is $\sum_{n=0}^\infty {(-1)^n \dfrac{2^{2n}}{(4n)!}z^{4n}}$ thus $n$ is doubled everywhere. Where is my error? I don't understand what I might have done wrong.
Where is the error? $$\frac{(i+1)^{2n} + (i-1)^{2n}}{2} = i^n2^n$$is false when $n$ is odd. (The left side is $0$.)