What would the equation to sum (shown below) of $ 3\cdot 10^{-n-1}$? Just like $ 2^x-1$ is the sum of $ 2^{x-1}$.
$$\sum_{n=0}^{\infty} 3\cdot 10^{-n-1}$$
This would be 0.3+0.03+0.003. . .
this would help me greatly in finding a limit for something.
UPDATE: I found (via calculator) it's $0.3021339806 \times 0.099570245^{x}$ but I'd like something smoother
You have that
$$0.3 + 0.03 + 0.003 + \cdots = \sum\limits_{n = 1}^\infty {\frac{3}{{{{10}^n}}}} $$
Now use that $$\sum\limits_{n = 1}^\infty {{a^n}} = \frac{a}{{1 - a}}$$
whenever $|a|<1$