Expression generating $\left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right)$

Question

I'm looking for a closed-form expression (in terms of $n$), that will give the sequence

$$ (s_n) = \left( \frac{3}{10}, \, \frac{3}{10} + \frac{33}{100}, \, \frac{3}{10} + \frac{33}{100} + \frac{333}{1000}, \dots \right). $$

Can anyone think of one? I made a related post to this question several minutes ago but I realized I was interpreting the sequence wrong.

Answer 1

$$s_n=\sum_{k=1}^n\frac{3\sum_{l=0}^{k-1}{10^l}}{10^k}.$$ Using the geometric sequence sum formula this simplifies considerably to: $$s_n=\frac{1}{27} (9n-1 + 10^{-n} ). $$

Answer 2

Built on my answer to the OP's previous post of the similar question, then:

$s_n = \displaystyle \sum_{k=1}^n \left(\dfrac{1}{3} - \dfrac{1}{3\cdot 10^k}\right)$

Answer 3

Just try to give another thought of expressing your series. $$ s_n=3\sum_{k=1}^n\frac{n+1-k}{10^{\large k}} $$