Question

Write the Maclaurin series expansion of the following functions:

cos²x

Answer 1

Let f(x) = cos²x

f(x) = cos²x, f(0) = 1

f'(x) = – 2 cos x sin x, f'(0) = 0

= – sin 2 x

f”(x) = – 2 cos 2x, f”(0) = – 2

f”‘(x) = 4 sin 2x, f”‘(0) = 0

f^IV(x) = 8 cos 2x, f^IV(0) = 8

f^V(x) = – 16 sin 2x, f^V(0) = 0

f^VI(x) = – 32 cos 2x, f^VI(0) = – 32

Maclaurin’s expansion is

f(x) = `sum_("n" - 0)^oo ("f"^(("n"))(0)x^"n")/("n"!)`

= `"f"(0) + ("f'"(0)x)/(1!) + ("f''"(0) x^2)/(2!) + ... + ("f"^(("n"))(0)x^"n")/("n"!) + ...`

cos²x = `1 + 0 - 2/(2!) x^2 + 0 + 8/(4!) x^4 + 0 - 32/(6!) x^6 + ...`

cos²x = `1 - (2x^2)/(2!) + (2^3x^4)/(4!) - (2^5x^6)/(6!) + ...`