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

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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 :)
 
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Answers and Replies

  • #2
HallsofIvy
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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 [itex]\sum_{n= 0}^\infty x^n[/itex]. And, to make "[itex]1/(1- x)[/itex]" look like "[itex]1/(1+x^2)[/itex]" replace x by [itex]-x^2[/itex], not just [itex]x^2[/itex].

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 :)
 
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  • #3
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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?
 
  • #4
Dick
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Yes. 1/(1+r)=sum (-r)^n for n=0 to infinity if |r|<1. Sub away.
 
  • #5
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Thanks so much, Dick :)
Unfortunately, the test still completely kicked my butt. Haha oh well.
 

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