MacLaurin Series Integration

  • Thread starter mateomy
  • Start date
  • #1
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MacLaurin Series Integration....

I have to find the MacLaurin for f(x)=ln(1+x^2)

So i started off by finding the derivative of the function getting

[tex]
\frac{2x}{1+x^2}
[/tex]

My issue lies with the 2x in the numerator. I know how to bring the x into the series, but the two? Do I leave it on the outside or do I bring it in and put it to the nth power? I think my brain is fried, I seem to be getting contradictory information from various sources. So, Im bringing it here and throwing it up on the board.

Should it look like this:

[tex]
\sum_{n=0}^{\infty} (-1)^n x^{2n+1} 2^n
[/tex]

or this:

[tex]
2 \sum_{n=0}^{\infty} (-1)^n x^{2n+1}
[/tex]

Thanks.
 

Answers and Replies

  • #2
307
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...and then I integrate it from there. I get that part, its just this one little step.
 
  • #3
HallsofIvy
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Homework Helper
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To find the MacLaurin series for
[tex]\frac{2x}{1+ x^2}[/tex]
Think of it as
[tex](2x)\left(\frac{1}{1-(-x^2)}[/tex]

Now, the sum of a geometric series is given by
[tex]\sum ar^n= \frac{a}{1- r}[/tex]
so think of this as a geometric series with [itex]r= -x^2[/itex] and a= 2x.
 

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