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The Taylor series of cosine is $$\cos(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}=\frac{x^0}{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}\mp\ ...$$ If we now plot the summands (ignoring the sign and the first summand for simplicity), one gets the following plot using GeoGebra: In this picture it seems as if the distance between the graphs is the same for the different summands. Furthermore, the distance seems to be something around $\frac{\pi}{4}$.

Is this a fact? And if so, why?

I'd guess that it has something to do with the fact that we can write Pi using the Leibniz-series: $$\frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}\ ...$$ However I can't quite see the connection between those two series.

It seems that this has very little to do with the cosine, but rather with monomials themselves - but why do even monomials divided by the factorial of their exponent have a distance in $x$-direction of $\frac{\pi}{4}$?