Using the finite sum formula evaluate the following sigman

Using the finite sum formula evaluate the following: sigma_n = 0^20 e^-jn pi/2 sigma_n = -3^1 e^jn pi/2 Using the infinite sum formula evaluate the following: sigma_n = 0^infinity (1/2)^n e^jn 3 pi/2 sigma_n = 1^infinity (1/3)^n e^jn pi/2 Evaluate the following: sigma_n = - infinity^infinity |e^-j pi n|

Solution

b) The given series is a geometric series with a common ratio e ^j/2 between successive terms. Also a = e^-3j/2 . Thus, as in part a) above, the sum of the given geometric series is e^-3j/2 ( 1 – e^4j/2 )/( 1 -e^j/2).

      

2. a) The formula for the sum of an infinite geometric series is a/(1 – r) where a and r are as above. Here a = (1/2)⁰ e^j*0*3/2 = 1*e⁰ = 1 and r = (1/2)e^3j/2 . Hence the sum of the given series is 1/[1 - (1/2)e^3j/2 ].

b) Here a = (1/3) e^j/2and r = (1/3)e^j/2.Hence the sum of the given series is[(1/3)e^j/2]/[1-(1/3) e^j/2]   

3 If j is positive, the 1^st term is |e^{-j (-)}| = |e^j |= and if j is negative, the last term is similarly |e-^j |=   . Hence the sum of the given series is .