Techniques to deriving DTFTs
As an example, the rules of series can work. A typical example is the $x[n] = a^n u[n]$ where $|a| &lt; 1$. For instance, say we want to find the DTFT of the signal $x[n] = 0.9^n u[n]$. Then,

$$X(e^{j\omega}) = \sum_{n = -\infty}^{\infty} 0.9^n u[n] e^{-j\omega n} \\
= \sum_{n = 0}^{\infty} (0.9 e^{-j\omega})^n$$

This is where the series work (geometric power series):

$$\sum_{n = 0}^{\infty} a^n = \frac{1}{1-a}$$

Then our DTFT is:

$$X(e^{j\omega}) = \frac{1}{1 - 0.9 e^{-j\omega}}$$