Calculus under the Algebra

Calculus Level 3

Simplify 1 ( 1 / 3 ) + ( 1 / 5 ) ( 1 / 7 ) . . . . π \frac{1-(1/3)+(1/5)-(1/7)....}{ \pi}


The answer is 0.25.

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1 solution

Paul Ryan Longhas
Feb 19, 2015

Since, t a n 1 x = x ( x 3 / 3 ) + ( x 5 / 5 ) . . . tan^{-1}x = x-(x^3/3)+(x^5/5)... for x 1 |x| \leq 1 = > t a n 1 1 = 1 ( 1 / 3 ) + ( 1 / 5 ) . . . => tan^{-1}1 = 1-(1/3)+(1/5)... But, t a n 1 1 = π 4 tan^{-1}1 = \frac{ \pi}{4} = > 1 ( 1 / 3 ) + ( 1 / 5 ) . . . = π 4 => 1-(1/3)+(1/5)... = \frac{ \pi}{4} Hence, = > ( π 4 ) / π = 1 4 = 0.25 => ( \frac{ \pi}{4})/ \pi = \frac{1}{4} = 0.25

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