Question

Write the Maclaurin series expansion of the following functions:

e^x

Answer 1

Let f(x) = e^x

f(x) = e^x f'(0) = e° = 1

f(x) = e^x f'(0) = e° = 1

f”(x) = e^x f”(0) = e° = 1

Maclaurin ‘s expansion is

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

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

∴ e^x = `1 + 1/(1!) x + 1/(2!) x^2 + ...`

e^x = `1 + x/(1!) + x^2/(2!) + ...`