I have this question from coursera tutorial. Howver, it has been a long time that I did nothing with maths and cannot solve this problem although I found the formula of geometric series. It is important to understand it. Therefore, I will appreciate a step by step solution and help.
Let $s[n]=\displaystyle \frac{1}{2^n}+j\frac{1}{3^n}$.Compute $\displaystyle\sum_{n=1}^{\infty}s[n]$.
Formula is as below but I couldn't manage to solve with complex numbers.
solution is as follows:
but I couldn't understand how 1/2 and 1/3 comes.
The sum of first $n$ terms of a geometrical series is given by $$ S_n = a \; \frac{1-z^{n}}{1-z} $$ where $a$ is the first term. For $n \rightarrow \infty$ $$ S_\infty = \frac{a}{1-z} $$ ONLY for $|z|<1$. Since your problem wants us to start the summation from $n=1$, rather than $n=0$, the first terms will be $1/2$ and $1/3$ respectively rather than $1$ and $1$.