Limit of Rational Expression with Repeating Integer I'm wondering if someone could point me in the right direction for the following problem:



I'm trying to express $(a + aa + \cdots + a\cdots a)/10^n$ as a summation that I can take the limit of as $n \to \infty$ and then use geometric series tools to get that down to something I can work with. However, I'm having quite a bit of trouble posing this as a summation. Should I be trying to express it recursively? It seems like that might get me somewhere. All of the things I've done so far have given me sums that diverge, which is clearly incorrect.
If I did not do any mistake, continuing Alex Becker's answer, the sum can be written as $$S(n)=\frac{a}{9}  \left(\frac{10}{9}- \left(\frac{10}{9}+n\right)10^{-n}\right)$$
Note that $99\cdots 9 = 10^k - 1$. Thus $aa\cdots a = \frac{a}{9}(10^k-1)$ and the sum is
$$\frac{a}{9}\sum_{k=1}^n \frac{10^k-1}{10^n}$$