Solution to simple recursive series

I'm working on some math programming and ran into the following recursive series.

$\displaystyle\sum\limits_{i=1}^n a_n$.
Where $a_{n} = Ca_{n-1}$ and $0 \leq C \leq 1$ and $a_0$ is given;

Because my constant values are always between 0 and 1, I know that the series converges and I can't help but thinking that there must be a solution.

My code looks something like

value = 1000000;

constant = 0.9999;
total = 0;
tolerance = 0.001;

while(value > tolerance)
{
    value = constant * value;
    total = total + value;
}

Problem is, as is the case with the initial values provided in the snippet above, the algorithm takes a long time to complete when $C$ approaches $1$

Answer

$a_k$ can be written as:

$$a_k = a_0 \cdot \underbrace{C \cdot C \cdots C}_{k \text{ times}} = a_0 C^{k}$$

Where $a_0 = 1000000$ and $C = 0.9999$. ($0 \lt C \lt 1$)

The sum is a geometric series. It has the following closed form:

$$
\sum_{k=0}^n a_k = a_0 \frac{1-C^{n+1}}{1-C}
$$

And as $n \to \infty$:

$$
\sum_{n=0}^\infty a_n = \frac{C_0}{1-C}
$$

On the other hand, if $C=1$, then $a_k=a_0$ and the sum becomes:

$$
\sum_{k=0}^n a_k = (n+1)a_k
$$

And this diverges as $n \to \infty$.