what is the taylor series ?

Question

what is the taylor series for cos(x^3)

Answer 1

Hello Matt! Good question; let me make this as simple as possible!

You may recall that the Maclaurian series (Taylor series at x = 0) for cosx is:

cosu ≈ 1 - (u ^ 2 / 2) + (u ^ 4 / 4!) - ..... ( u ^ (2n) / (2n!) ).....

Now, if we set u = x ^ 3 (this is substitution here) and put that back into the equation, we get:

cos(x^3) ≈ 1 - (x ^ 6 / 2) + (x ^ 12 / 4!) - ..... ( x ^ (6n) / (2n!) ).....

I hoped that helped! I did this under the assumption that it was centered at x = 0. If you have other questions for me, please let me know and I'll be glad to help out!