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Oct 22nd, 2017

(C)

Z7-z9+z11-…

z7 sum (-1)n z2n  from 0 to infinity = z7/(1+z2)

i = cos pi/2+ i sin ppi/2

i(2n+3)i =exp( i pi/2)(2n+3)i = exp(-(2n+3)pi/2) = e-(2n+3)pi/2

| (-1)ni(2n+3)i | =| e-(2n+3)pi/2 |

Zn+1/zn  = e-(2n+5)pi/2+(2n+3)pi/2= e-pi = 1/epi < 1

Hence the series is convergent

Sum (-1)n e-(2n+3)pi/2 from 2 to infinity

= e-3pi/2 sum (-1)n (e-pi)n from 2 to infinity

= e-7pi/2 sum (-1)n(epi)n from 0 to infinity

e-7pi/2 /(1+epi)


Dec 24th, 2014

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