Power Series: Find First 4 Terms of Series Expansion for Sec x

[Fermatslast]

Can anyone help me with this? It's been really annoying me and I think I am just forgetting something:

Using the series expansion for cosx in powers of x find the first four non-zero terms of the corresponding series for sec x.

I get obviously that as secx is 1/cosx it is a case of division of power series but I get confused because it is 1/a power series rather than a power series/a power series so I'm not sure how to treat this.

I would really appreciate help.

 

[Tide]

Basically you want to expand this:

$$\frac {1}{1-stuff}$$

which you can expand into

$$1 + stuff + stuff^2 + stuff^3 + O(stuff^4)$$

and you'll do a lot of algebra to expand the powers of stuff all of which must contain several terms in powers of x. Enjoy!

 

[Fermatslast]

Thankyou

Thankyou very much. I will give it a go :)

 

[josephcollins]

Okay, get the maclaurin expansion for cosx

namely, 1-x^2/2 + x^4/4! -...

Then write (1+(x^4/4! -x^2/2+...) for cos x

Since sec x =(cosx)^-1 you can now use the binomial theorem to deduce the series for secx

Regards,



Joe