Question

Write the Maclaurin series expansion of the following functions:

cos x

Answer 1

Let f(x) = cos x

f(x) = cos x, f(0) = 1

f'(x) = – sin x, f'(0) = 0

f”(x) = – cos x, f”(0) = –1

f”'(x) = sin x, f”'(0) = 0

f^IV(x) = cos x, f^IV(0) = 1

f^V(x) = – sin x, f^V(0) = 0

f^VI(x) = – cos x, f^VI(0) = –1

Maclaurin ‘s expansion is

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

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

∴ cos x = `1 + 0 - 1/(2!) x^2 + 0 + 1/(4!) x^4 + 0 - 1/(6!) x^6 + ...`

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