chatterclaw73
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Express r/(r-1)(r+1) in partial fractions, and use your results to sum the series
2/(1*3) - 4/(3*5) -..... +((-1)^n+1*2n)/(2n-1)(2n+1)

This is the last question of an exercise following an explanation of the method of difference approach.
I have the partial fractions as
1/(2r+1) +1/2(r-1) but I cannot see how to apply the method. I do not have any working to show as I am at a loss how to begin. Thanks in advance for any advice or pointers.
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RDKGames
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(Original post by chatterclaw73)
Express r/(r-1)(r+1) in partial fractions, and use your results to sum the series
2/(1*3) - 4/(3*5) -..... +((-1)^n+1*2n)/(2n-1)(2n+1)

This is the last question of an exercise following an explanation of the method of difference approach.
I have the partial fractions as
1/(2r+1) +1/2(r-1) but I cannot see how to apply the method. I do not have any working to show as I am at a loss how to begin. Thanks in advance for any advice or pointers.
The general term is  \displaystyle (-1)^{k+1} \dfrac{2k}{(2k-1)(2k+1)}

whose sum is from k=1 up to k=n.

Use the sub r=2k in your partial frac and you reduce down to what you need.
Last edited by RDKGames; 1 month ago
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chatterclaw73
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(Original post by RDKGames)
The general term is  \displaystyle (-1)^{k+1} \dfrac{2k}{(2k-1)(2k+1)}

whose sum is from k=1 up to k=n.

Use the sub r=2k in your partial frac and you reduce down to what you need.
The partial fraction has a positive sign. I thought that the cancelling of terms depended on a negative sign. Sorry if I am being dim.
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mqb2766
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(Original post by chatterclaw73)
The partial fraction has a positive sign. I thought that the cancelling of terms depended on a negative sign. Sorry if I am being dim.
The partial fractions for the first term have two positive signs
The partial fractions for the second term have two negative signs
... and so on ... Just write down the first few terms.
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chatterclaw73
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(Original post by mqb2766)
The partial fractions for the first term have two positive signs
The partial fractions for the second term have two negative signs
... and so on ... Just write down the first few terms.
Thank you. Can you recommend any books or websites for me on summing series. I need to practice more.
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mqb2766
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(Original post by chatterclaw73)
Thank you. Can you recommend any books or websites for me on summing series. I need to practice more.
A quick google
https://www.examsolutions.net/tutori...f-differences/
https://www.revisely.co.uk/alevel/fu...cel/corepureqs
http://mathsathawthorn.pbworks.com/f...nces%2Bpdf.pdf
https://www.furthermathstutor.co.uk/fp2/series.html
Last edited by mqb2766; 1 month ago
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chatterclaw73
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Thank you. Much appreciated.
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