What power series represents 1/(1+x^2) on (-1,1)

[CanaryK]

Homework Statement

Word for word, the book says "Find a power series that represents 1/(1+x^2) on (-1, 1)

Homework Equations

It's in the chapter that talks about power series, so I think they want me to use the fact that 1/(1-x) is a power series with a=1 and r=x, but if I just substitute in -x^2 for x, that makes chain rule issues. The chapter also talks about term-by-term integration, which confuses me more than little bit.

The Attempt at a Solution

Uhm... I don't really have much yet. I think this is probably an extremely basic question and I'm just missing some underlying technique or trick to solve it.
I know that the antiderivative of 1/(1+x^2) is arctan(x) of but we don't know the series for arctan(x), so that doesn't really help. The derivative of 1/(1+x^2) is -2x/(1+x^2)^2, which doesn't seem to help either.

Thanks for any help :)

 

[HallsofIvy]

Homework Statement

Word for word, the book says "Find a power series that represents 1/(1+x^2) on (-1, 1)

Homework Equations

It's in the chapter that talks about power series, so I think they want me to use the fact that 1/(1-x) is a power series with a=1 and r=x, but if I just substitute in x^2 for x, that makes power series issues.

What do you mean by "power series issues"? Of course, 1/(1-x), itself, is not a "power series", it is the sum of the geometric series \sum_{n= 0}^\infty x^n. And, to make "1/(1- x)" look like "1/(1+x^2)" replace x by -x^2, not just x^2.

The chapter also talks about term-by-term integration, which confuses me more than little bit.

The Attempt at a Solution

Uhm... I don't really have much yet. I think this is probably an extremely basic question and I'm just missing some underlying technique or trick to solve it.
I know that the antiderivative of 1/(1+x^2) is arctan(x) of but we don't know the series for arctan(x), so that doesn't really help. The derivative of 1/(1+x^2) is -2x/(1+x^2)^2, which doesn't seem to help either.

Thanks for any help :)

 

[CanaryK]

Wow I'm sorry, for "power series issues" I actually meant to type "chain rule issues". Not sure what I was thinking there haha! Multiple bad typos in that, I'll go back and fix them in a second.

I'm not quite understanding your answer - so I CAN just go back and sub in -x^2 (that's what I meant, I swear haha)? And it doesn't matter that x^2 is a different order from x?

 

[Dick]

Yes. 1/(1+r)=sum (-r)^n for n=0 to infinity if |r|<1. Sub away.

 

[CanaryK]

Thanks so much, Dick :)
Unfortunately, the test still completely kicked my butt. Haha oh well.