Power or Energy Signal?
The energy of a discrete-time signal is defined as

$$E_x=\sum_{n=-\infty}^{\infty}|x(n)|^2\tag{1}$$

and its power is given by

$$P_x=\lim_{N\rightarrow\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}|x(n)|^2$$

With $x(n) = e^{j (n \pi/2 + \pi/8) }$ we have $|x(n)|^2=1$ which implies that the sum in (1) does not converge, i.e. $x(n)$ has infinite energy. For the power we get

$$P_x=\lim_{N\rightarrow\infty}\frac{2N+1}{2N+1}=1$$

The signal $x(n)$ has finite power and is consequently a "power signal".