finite expansion of order 2 in the neighborhood of 0 of (cosx - 1)/x2

Question

I need the steps plzzz

Answer 1

The expansion of cos(x), using the Maclaurin series, is given by,

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ... + x^n.cos(n.pi/2)/n!

cos(x) - 1 = - x^2/2! + x^4/4! - x^6/6! + ... + x^n.cos(n.pi/2)/n!

(cos(x) - 1)/x^2 = -1/2! + x^2/4! - x^4/6! + ... + x^(n-2).cos(n.pi/2)/n!

Taken to the 2nd order, the expression becomes,

(cos(x) - 1)/x^2 = -1/2! + x^2/4!