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QUESTION


Find the Taylor series for $\cos z$ about $z=\pi/2.$



ANSWER


Put $w=z-\pi/2$. Then $\cos z=\cos(w+\pi/2)$. Put
$g(w)=\cos(w+\pi/2)=g(0)+wg^{'}(0)+{w^2\over 2!}g^{''}(0)+\cdots$.
Now $g(0)=0,$, $g^{'}(0)=-1\cdots$ so $$\cos
z=-(z-\pi/2)+(z-\pi/2)^3/3!+\cdots$$




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