1 What is the sum of the seriesSolutionapi2 pi386 pi532120 p

1. What is the sum of the series?

Solution

a)(pi/2) -(pi³)/(8*6) +(pi⁵)/(32*120)+...

=(pi/2)-(pi/2)³/3! +(pi/2)⁵/(5!)-......

=sin(pi/2)

=1

b)(1/3) -1/(3*3*3) +1/(9*3 *5) -1/(27*3 *7) +1/(81*3 *9) -...

=(1/3)[1 -(1/3)/3 +(1/9)/5 -(1/27)/7 +(1/81)/9 -..........]

=(1/3)[(1/3)⁰/(2*0 +1) -(1/3)/(2*1 +1) +(1/3)²/(2*2 +1) -(1/3)³/(2*3 +1) +(1/3)⁴/(2*4 +1) -..........]

=(1/3)[tan^-1((1/3))]/(1/3)

=[tan^-1((1/3))]

=pi/6