## Yahoo! one year ago $\lim_{n \rightarrow \infty}(\frac{ 1 }{ 1*3 }+\frac{ 1 }{ 3*5 }+......\frac{ 1 }{ (2n-1)(2n+1) })$

1. hartnn

2=3-1 so, 1/1*3 = 1/2(2/1*3) = 1/2((3-1)/3*1) = 1/2[1-1/3] do this for every term. should i do it in latex, or you got it ?

2. Yahoo!

Latex...Plzz

3. hartnn

$\frac{1}{1 \times 3}=\frac{1}{2} \times \frac{2}{1 \times 3}=\frac{1}{2} \times[\frac{3-1}{1 \times 3}]=\frac{1}{2} \times[\frac{1}{1 }-\frac{1}{ 3}]$ do this for every term.

4. hartnn

2nd term = 1/3 -1/5 notice 1/3 will get cancelled, and if you go on, all the terms excepts 1st and last will get cancelled.

5. Yahoo!

Would u Call This Partial Fraction Decomposition?

6. hartnn

yes. i would.

7. Yahoo!

it would be n /(2n+1) ryt? @hartnn

8. hartnn

yes.

9. hartnn

then limit=... ?

10. Yahoo!

1/2

11. hartnn

correct.

12. Yahoo!

Thxxx

use the telescopic's principle, it can be 1/2 (1 - 1/(2n+1)) 1/2 (2n/(2n+1)) just look the cofficient of n, they are same) :)

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