Final answer:
The power series for tan(x) as the term x⁶, divide the power series for sine by the power series for cosine and expand the expression and we get tan(x) = x + (x^3)/3 + (2x^5)/15 + (17x^7)/315 + ....
Step-by-step explanation:
To find the power series for tan(x) as the term x⁶, we can start with the power series for sine and cosine:
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...
cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...
Then, we can use the identity tan(x) = sin(x)/cos(x) to divide the power series for sine by the power series for cosine:
tan(x) = (x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...) / (1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...)
Now, we can expand the expressions and collect like terms:
tan(x) = x + (x^3)/3 + (2x^5)/15 + (17x^7)/315 + ...