Question

Find the expansion of cos x about the point x=0

Answer 1

Cos x = 1 -
`(x^2)/(2!)` +
`(x^4)/(4!)` -
`(x^6)/(1!)` + ...

Explanation:

We use Taylor series expansion to answer this question.

We have to find the expansion of cos x at x = 0

f(x) = cos x, f'(x) = -sin x, f''(x) = -cos x, f'''(x) = sin x, f''''(x) = cos x

Now we evaluate them at x = 0.

f(0) = 1, f'(0) = 0, f''(0) = -1, f'''(0) = 0, f''''(0) = 1

Now, by Taylor series expansion we have

f(x) = f(a) + f'(a)(x-a) +
`(f''(a)(x-a)^2)/(2!)` +
`(f'''(a)(x-a)^3)/(3!)` +
`(f''''(a)(x-a)^4)/(4!)` + ...

Putting a = 0 and all the values from above in the expansion, we get,

Cos x = 1 -
`(x^2)/(2!)` +
`(x^4)/(4!)` -
`(x^6)/(1!)` + ...