Super Simple Series!!

Algebra Level 3

If 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + . . . . . . = x 4 1-\frac { 1 }{ 3 } +\frac { 1 }{ 5 } -\frac { 1 }{ 7 } +\frac { 1 }{ 9 } -\frac { 1 }{ 11 } +......=\frac { x }{ 4 }

then the value of 1 1.3 + 1 5.7 + 1 9.11 + . . . . . . . \frac { 1 }{ 1.3 } +\frac { 1 }{ 5.7 } +\frac { 1 }{ 9.11 } +....... is

x 8 \frac { x }{ 8 } x 4 \frac { x }{ 4 } x 16 \frac { x }{ 16 } x 6 \frac { x }{ 6 }

Krishna Ramesh by Krishna Ramesh , 22, India

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2 solutions

Jaydee Lucero
Sep 21, 2014

Hint : Partial fraction decomposition. For example, 1 1 ⋅ 3 = 1 2 ( 1 1 − 1 3 ) \frac{1}{1 \cdot 3}=\frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}\right) From there you will notice something cool. :)

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Ronak Agarwal
Sep 21, 2014

It is worthy to note that x = π x=\pi

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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