Division of complex power series

[Pyroadept]

Homework Statement

Find tan(z) up to the z^7 term, where tan(z) = sin(z)/cos(z)

Homework Equations

sin(z) = z - z^3/3! + z^5/5! - z^7/7! + ...

cos(z) = 1 - z^2/2! + z^4/4! - z^6/6! + ...

The Attempt at a Solution

Hi,
Seeing as sin and cos have the same power series as for when they are real, can you just divide the complex polynomials?

i.e. (z - z^3/3! + z^5/5! - z^7/7! + ...) / (1 - z^2/2! + z^4/4! - z^6/6! + ...) = z + z^3/3 + 2z^5/15 + 17z^7/315 + ...

which is tan(z)? (Assuming it has the same complex power series as real power series, considering sin and cos do?)

Thanks for any help

 

[Dick]

Yes, you can do it that way.