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*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(\omega) + 1/(j\omega))\$. 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^{j \omega t})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 \$ \infty \$ as my answer, unless of course the integration is wrong.

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