What is the power series for cos(u^2) up to the u^24 term?

[iceman]

Hello, can anyone help me on this one? I am finding problem pretty tricky indeed.

I know the power series (or Taylor series) for cosx is

1-x^2/2!+x^4/4!-x^6/6!+... (It is convergent for all element x is a member of set R)

Q) Write down the power series for cos(u^2) up to the u^24 term.

Your help is kindly appreciated.

 

[selfAdjoint]

First write the cos(x) series you know. Then plug in u^2 for each x in the series. What will that do to the powers of u? What will it do to the coefficients? After you answer those questions try this. What will be the factorial in the denominator when the power of u is 24?

 

[tacman]

Hi there! The power series for cos(u^2) up to the u^24 term would be:

1 - (u^2)^2/2! + (u^2)^4/4! - (u^2)^6/6! + (u^2)^8/8! - (u^2)^10/10! + (u^2)^12/12! - (u^2)^14/14! + (u^2)^16/16! - (u^2)^18/18! + (u^2)^20/20! - (u^2)^22/22! + (u^2)^24/24!

This is a bit tricky because we have to take into account that u is being squared. So, we have to raise each term to the power of 2. For example, (u^2)^2 becomes u^4, (u^2)^4 becomes u^8, and so on. Then, we divide each term by the corresponding factorial value, just like in the power series for cosx. I hope this helps! Let me know if you have any other questions.