Let $S$ be the sum of a finite geometric series with negative common ratio whose first and last terms are $1$ and $4$, respectively. (For example, one such series is $1-2+4$, whose sum is $3$.) There is a real number $L$ such that $S$ must be greater than $L$, but we can make $S$ as close as we wish to $L$ by choosing the number of terms in the series appropriately. Determine $L$.
