Maclaurin series

Question

Find the first three nonzero terms of the Maclaurin series of f(x)=arctan(x)

Answer 1

Find the first three nonzero terms of the Maclaurin series of f(x)=arctan(x)

So the McLaurin series of a function f(x) is as follows:

f(x) = f(0) + xf'(0) + 1/2 *x^2 * f''(0) + 1/3! * x^3 f'''(0) + ...

Note that if f(x) = arctan(x) then to get the first three terms we want to find

f(0) = arctan(0)

= 0

(this doesn't count as a term since it's zero)

f'(x) = 1/(1+x^2)

f'(0) = 1

f''(x) = -2x/(1+x^2)^2

f''(0) = 0

(this doesn't count as a term since it's zero)

f'''(x) = (6x^2 - 2)/(1+x^2)^3

f'''(0) = -2

f^(4)(x) = -24x(x^2-1)/(1+x^2)^4

f^(4)(0) = 0

(this doesn't count as a term since it's zero)

f^(5)(x) = 24(5x^4-10x^2 + 1)/(1+x^2)^5

f^(5)(0) = 24

Whew! Okay, so the first, third and fifth derivative values are non-zero and help make up the first three terms of the McLaurin expansion of arctan(x)

We get

arctan(x) ≈ x*1 + 1/3! * x^3 * (-2) + 1/5! * x^5 * (24)

= x - 1/3 * x^3 + 1/5 * x^5