maclaurin series

Question

Find the first three nonzero terms in the Maclaurin series of f(x)= tan x.

Answer 1

Maclaurin or Taylor series for f(x) = tanx

= the series for sinx divided by the series for cosx

tanx = x + x^3/3 + 2x^5/15 +17x^7/315 + ...

1st 3 terms = x + (1/3)x^3 + (2/15)x^5 + ...

or

0 + x + 0 = x for 1st 3 terms

but 1st 3 non-zero terms are:

tanx = f(0) + f'(0)x/1! + f"(0)x^2/2! + fiv(x^3)/3! + fv(x^4)/4! + fvi(x^5)5! + ...

= 0 + x + 0 + x^3/3 + 0 + 2x^5/15 + ...

= x + x^3/3 + 2x^5/15 + ...

= x +x^3/3 + 2x^5/15 + ...

tanx)' = 1+tan^2(x)

(tanx)" = 2tanx +2tan^3(x)

(tanx)"' = 2+ 8tan^2(x) + 6tan^4(x)

(tanx)iv =16tanx + 40tan^3(x) +246tan^4(x)

(tanx)v = 16+ 136tan^2(x) +240tan^4(x) = 120tan^6(x)

all even derivatives = 0 when x=0

only the odd derivatives are non zero terms

it helps to review derivatives of trig functions