Is the sum of the series $\sum 1/3^n$ equal to $1/2$ or $3/2$?

Question

The formula for a geometric series as I know it is $\sum ar^{n-1}$, where $r$ is the common ratio and $a$ is the first number the common ratio is multiplied with.

If we're to conform to that formula, then the series $\sum 1/3^n$ would be $\sum \frac13 \frac{1}{3^{n-1}}$, which would tell you that $a$ = $r$ =$1/3$. From there, using the formula for the sum of a geometric series, $a/1-r$, gives you $1/2$.

However, the sum is apparently equal to $3/2$, which the formula for the sum would only give if you took $a$ to be $1$ instead of $1/3$.

So why is $a$ apparently $1$, despite the fact that I conformed to the formula properly?

Any help is appreciated.

Answer 1

Depends if you start the index at $0$ or $1$

Answer 2

$$ \frac{1}{1-x}=\sum_{k=0}^{\infty}x^k$$ $$\frac{x}{1-x}=\sum_{k=1}^{\infty}x^k$$