Obtain-a-maclaurin-expansion-for-a-e-cos-x-b-e-cos-2-x-

Question Number 83296 by Rio Michael last updated on 29/Feb/20

Obtain a maclaurin expansion for

a) e^{\cos x} b) e^{\cos^{2} x}

Commented by niroj last updated on 29/Feb/20

(a) . e^{\cos x}

let, f (x) = e^{\cos x}, f_{0} (0) = e^{\cos 0} = e

f_{1} (x) = e^{\cos x} . (- sinx), f_{1} (0) = 0

f_{2} (x) = \sin^{2} x . e^{\cos x} - \cos x . e^{\cos x}

f_{2} (0) = - e

f_{3} (x) = 2 \sin x \cos x . e^{\cos x} - \sin^{2} x . e^{\cos x} . sinx + \sin x . e^{\cos x} + \cos x . e^{\cos x} . \sin x

f_{3} (0) = 0

we know {maclaurine}^{'} s series . .

f (x) = f_{0} (0) + \frac{x}{1!} f_{1} (0) + \frac{x^{2}}{2!} f_{2} (0) + \frac{x^{3}}{3!} f_{3} (0) + \dots

e^{\cos x} = e + \frac{x}{1 \times 1} (0) + \frac{x^{2}}{2 \times 1} (- e) + \frac{x^{3}}{3 \times 2 \times 1} (0) + \dots

e^{\cos x} = e - x^{2} e + \dots .

Commented by Rio Michael last updated on 29/Feb/20

t h a n k s s i r