(C)

Z^{7}-z^{9}+z^{11}-…

z^{7} sum (-1)^{n} z^{2n } from 0 to infinity = z^{7}/(1+z^{2})

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)^{n}i^{(2n+3)i} | =| e^{-(2n+3)pi/2} |

Z_{n+1}/z_{n } = e^{-(2n+5)pi/2+(2n+3)pi/2}= e^{-pi} = 1/e^{pi} < 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}(e^{pi})^{n} from 0 to infinity

e^{-7pi/2} /(1+e^{pi})

Secure Information

Content will be erased after question is completed.

Enter the email address associated with your account, and we will email you a link to reset your password.

Forgot your password?

Sign Up