Thread: Taylor/Maclaurin series

Hello, I was wondering if someone could explain this problem:

Expand f(x)=x^3/(1+x^3) in powers of x for abs(x)<1.

I think it's a maclaurin expansion, but I've evaluated up to the 9th derivative at f(0), and I come up with 2 terms. How long do I have to expand it? or am I just doing it wrong.

Thanks.

Originally Posted by hasbro

Hello, I was wondering if someone could explain this problem:

Expand f(x)=x^3/(1+x^3) in powers of x for abs(x)<1.

I think it's a maclaurin expansion, but I've evaluated up to the 9th derivative at f(0), and I come up with 2 terms. How long do I have to expand it? or am I just doing it wrong.

Thanks.

There's an easier way to do this. First



Then from the geometric power series



replace the with giving




so putting these together gives

Thank you that's a lot easier.

What would you do if the function was (x^3)/(1+(x^4))?

Originally Posted by hasbro

Thank you that's a lot easier.

What would you do if the function was (x^3)/(1+(x^4))?

Use

and multiply by (even easier).

Indeed, you could have done in exactly that (easier) way:
and so is equal to the geometric series and then multiply each term by
where, in the last sum, j= n+3.