Telescoping Method & Partial Fractions PLEASE HELP

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Telescoping Method & Partial Fractions...PLEASE HELP!!!

Homework Statement


Find the sum of the series from n=1 to infinity...
2/(4n^2-1)


Homework Equations





The Attempt at a Solution


I want to use the telescoping method....
2/(4n^2) = 2/[(2n-2) * (2n+1)]

I am following an example in my book and this is where I don't know what they did next...but they got
An = 1/(2n-1) - 1/(2n+1)
 

Answers and Replies

  • #2
Tom Mattson
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The Attempt at a Solution


I want to use the telescoping method....
2/(4n^2) = 2/[(2n-2) * (2n+1)]
Correct. And how do you write down the partial fraction decomposition when your denominator has two nonrepeated, linear factors?
 
  • #3
tiny-tim
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2/(4n^2-1)...
2/(4n^2) = 2/[(2n-2) * (2n+1)]

An = 1/(2n-1) - 1/(2n+1)
Hi BuBbLeS01! :smile:

I'm confused … 1/(2n - 1) - 1/(2n + 1) is 2/(4n² - 1). :confused:

Next step … what is An + An+2 ? :smile:
 
  • #4
Tom Mattson
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Hi BuBbLeS01! :smile:

I'm confused … 1/(2n - 1) - 1/(2n + 1) is 2/(4n² - 1). :confused:
Right, he's asking how the author got that result. So he needs to revisit partial fraction decomposition.
 
  • #5
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Correct. And how do you write down the partial fraction decomposition when your denominator has two nonrepeated, linear factors?
2/(4n^2) = 2/[(2n-1) * (2n+1)]
So the LCD for the right side would be (2n-1)*(2n+1)

2(2n+1)/(2n-1) + 2(2n-1)/(2n+1)

For some reason that doesn't look right though??
 
  • #6
Tom Mattson
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2/(4n^2) = 2/[(2n-1) * (2n+1)]
So the LCD for the right side would be (2n-1)*(2n+1)
Hold on a second, you dropped a "-1" somewhere along the way. It should be this:

[tex]\frac{2}{4n^2-1}=\frac{2}{(2n-1)(2n+1)}[/tex]

2(2n+1)/(2n-1) + 2(2n-1)/(2n+1)

For some reason that doesn't look right though??
That's because it's not right, and I don't understand why you're writing it. What you're supposed to do is this:

[tex]\frac{2}{(2n-1)(2n+1)}=\frac{A}{2n-1}+\frac{B}{2n+1}[/tex]

That ought to jog your memory enough to finish.
 
  • #7
tiny-tim
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Hi BuBbLeS01!

That's because you've put a + in the middle instead of a - . :smile:
 
  • #8
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So is it just....
2 = A(2n+1) + B(2n-1)
 
  • #9
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A = 1 and B = -1
 
  • #10
tiny-tim
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:smile: Yes! :smile:
 
  • #11
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WOO-HOO LOL Thank you....I think I got it from here :)
 

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