Question

Write the Maclaurin series expansion of the following functions:

sin x

Answer 1

Let f(x) = sin x

f(x) = sin x, f(0) = 0

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

f”(x) = – sin x, f”(0) = 0

f”‘(x) = – cos x, f”'(0) = –1

f^IV(x) = sin x, f^IV(0) = 0

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

f^VI(x) = – sin x, f^VI(0) = 0

f^VII(x) = – cos x, f^VII(0) = –1

Maclaurin ‘s expansion is

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

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

∴ sin x = `0 + 1/(1!) x + 0 - 1/(3!) x^3 + 0 + 1/(5!) x^5 + 0 - 1/(7!) x^7 + ....`

sin x = `x/(1!) - x^3/(3!) + x^5/(5!) - x^7/(7!) + ...`