Find the function f (z) = cosz given in a Taylor series around -1. Specify the maximum disk where this representation is valid
Thanks for your help
Thanks for your help
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cos z
= cos((z - 1) + 1)
= cos(z - 1) cos 1 - sin(z - 1) sin 1
= (cos 1) * Σ(n = 0 to ∞) (-1)^n (z - 1)^(2n)/(2n)! - (sin 1) * Σ(n = 0 to ∞) (-1)^n (z - 1)^(2n+1)/(2n+1)!
Since both series converge for all z, the resulting series for cos z also converges for all z.
(The radius of convergence is infinite.)
I hope this helps!
= cos((z - 1) + 1)
= cos(z - 1) cos 1 - sin(z - 1) sin 1
= (cos 1) * Σ(n = 0 to ∞) (-1)^n (z - 1)^(2n)/(2n)! - (sin 1) * Σ(n = 0 to ∞) (-1)^n (z - 1)^(2n+1)/(2n+1)!
Since both series converge for all z, the resulting series for cos z also converges for all z.
(The radius of convergence is infinite.)
I hope this helps!