I'm working in a signals class for continuous signals, and we have this problem shown above. I have tried using this function f_1 X f_2 = F_1 * F_2, where I'm assuming this means multiplication of two functions is equal to the convolution of their fourier transforms. I'm using f_1 = 0.5^n and f_2 = u(n).

So I can calculate the fourier transorm of u(n) fine. It is (pi)(delta)(w) + 1/(jw)). However, I cannot for the life of me figure out 0.5^n. I tried to put it into the fourier transform integral integral of (0.5^t)/(e^(jwt))dt from negative infinity to infinity, but I end up with 0.5t/(e^(jw)), and when evaluated from negative infinity to infinity, I end up with infinity as my answer, unless of course the integration is wrong.

Therefore, either the answer is infinity * (pi)(delta)(w) + 1/(jw), which when convoluted would equal just the second function..? OR am I going about this problem completely wrong?