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i am trying to calculate the sum of this infinite series after having read the series chapter of my textbook: $$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}$$

my steps:

$$\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=\sum_{n=1}^{\infty}\frac{2}{4n^2-1}-\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=..help..=sum$$

i am lacking some important properties, i feel i am coming to the right step and cannot spit that out..

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Hint: Write your rational functions as a linear combination of fractions with denominator being a linear function. –  Ofir Dec 26 '12 at 10:47
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@doniyor taking $\displaystyle \lim_{n\to \infty} \left(\frac{1}{2} - \frac{1}{4n+2}\right)$ in the result posted in Elias's link you get $\frac{1}{2}$ –  Rustyn Dec 26 '12 at 10:55

4 Answers 4

up vote 3 down vote accepted

Hint: Partial Fraction decomposition:$$\frac{1}{4n^2-1}=\frac{1}{(2n-1)(2n+1)}=\frac12[\frac{1}{2n-1}-\frac{1}{2n+1}]$$ You must then compute the closed form of $$\sum_{n=1}^k[\frac{1}{2n-1}-\frac{1}{2n+1}]$$ Can you do that? Note that $$\sum_{n=1}^k\frac{1}{2n-1}=\frac11+\frac13+...+\frac1{2k-1}=\frac1{2\cdot 0+1}+\frac1{2\cdot 1+1}+...+\frac1{2(k-1)+1}=\sum_{n=0}^{k-1}\frac{1}{2n+1}=\sum_{n=1}^{k}\frac1{2n+1}+\frac{1}{2\cdot 0+1}-\frac1{2k+1}$$

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can you pls expand this step? $\sum_{n=1}^k\frac{1}{2n-1}=\sum_{n=0}^{k-1}\frac{1}{2n+1}$ –  doniyor Dec 26 '12 at 11:29
    
@doniyor Sure I can. –  Nameless Dec 26 '12 at 11:29

Note $\frac{1}{4n^2-1}=\frac{1}{(2n+1)(2n-1)}={\frac{1}{2}}\times\frac{(2n+1)-(2n-1)}{(2n+1)(2n-1)}={\frac{1}{2}}\times[\frac{1}{2n-1}-\frac{1}{2n+1}]$ for $n\in\mathbb N$

Let for $k\in\mathbb N,$ $S_k=\sum_{n=1}^{k}\frac{1}{4n^2-1}$ $\implies S_k={\frac{1}{2}}\sum_{n=1}^{k}[\frac{1}{2n-1}-\frac{1}{2n+1}].$ Thus for $k=1,2,...$

$S_1={\frac{1}{2}}\sum_{n=1}^{1}[\frac{1}{2n-1}-\frac{1}{2n+1}]=\frac{1}{2}(1-\frac{1}{3})$

$S_2={\frac{1}{2}}\sum_{n=1}^{2}[\frac{1}{2n-1}-\frac{1}{2n+1}]=\frac{1}{2}[(1-\frac{1}{3})+(\frac{1}{3}-\frac{1}{5})]=\frac{1}{2}(1-\frac{1}{5})$

$S_3={\frac{1}{2}}\sum_{n=1}^{3}[\frac{1}{2n-1}-\frac{1}{2n+1}]=\frac{1}{2}[(1-\frac{1}{3})+(\frac{1}{3}-\frac{1}{5})+(\frac{1}{5}-\frac{1}{7})]=\frac{1}{2}(1-\frac{1}{7})$

...

$S_k=\frac{1}{2}(1-\frac{1}{2k+1})$

$\implies\sum_{n=1}^{\infty}\frac{1}{4n^2-1}=\lim_{k\to\infty}S_k=\frac{1}{2}.$

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great, thanks Sugata –  doniyor Dec 26 '12 at 13:39
    
You're most welcome. –  Sugata Adhya Dec 26 '12 at 13:41
    
+1 nice answer. –  B. S. Dec 26 '12 at 16:13

Or we want to compute it fastly and use the formula $$\sum_{k=1}^\infty \frac1{k^2-x^2}=\frac1{2x^2}-\frac{\pi\cot\,\pi x}{2x}$$ where $x=\frac{1}{2}$ because $$\sum_{k=1}^{\infty}\frac{1}{4k^2-1}=\frac{1}{4}\sum_{k=1}^{\infty}\frac{1}{k^2-\left(\frac{1}{2}\right)^2}$$ Here you may find more information about this precious way.

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thanks, great help –  doniyor Dec 26 '12 at 20:51
    
@doniyor: you're welcome! –  Chris's sis Dec 26 '12 at 20:52

Hint: Work on $S_n=\sum_{k=1}^n\frac{1}{4k^2-1}$ and take its limit when $n\to\infty$. Note that $$\frac{1}{4n^2-1}=\frac{1}{2(2n-1)}-\frac{1}{2(2n+1)}$$

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Why don't you experts leave such problems for beginners like us ? :( –  Sugata Adhya Dec 26 '12 at 13:40
    
@SugataAdhya: Dear Sugata, I am not an expert. I am a teacher and like to help others in Maths. That's it. We are here to help eachother in learning Maths well. :-) –  B. S. Dec 26 '12 at 16:13
    
:), yeah, but you guys are doing wonderful job here.. thanks again for all you –  doniyor Dec 26 '12 at 20:50
    
Yes, as dear doniyor says, you're doing a wonderful job here! + –  amWhy Mar 2 '13 at 2:45

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