Connection to Normal distribution I've been working on finding the probability for the event, that the sum of $n$ independent random variables are less than $s$, when they are evenly distributed on $[0,1)$.
I've used the law of total probability to derive the formula:
$P(S<s) = \frac{1}{n!}\sum_{k=0}^{\lfloor s\rfloor}(-1)^k\binom{n}{k}(s-k)^n$
$f_S(s) = \frac{1}{(n-1)!}\sum_{k=0}^{\lfloor s\rfloor}(-1)^k\binom{n}{k}(s-k)^{n-1}$
Now I'm not very strong in probability, but I believe you could express the same thing using the normal distribution. Or at least approximate it. Is that so? In that case how would you express it, and how would you normally determine the approximation error?

For an explicit expression for the denisty function see MathWorld: Uniform Sum Distribution, for example. 
EDIT: 
The density function $f_n$ of the sum $U_1 + \cdots + U_n$, where $U_i$ are independent uniform$(0,1)$ variables, is given, according to the MathWorld link above, by
$$
f_n (x) = \frac{1}{{2(n - 1)!}}\sum\limits_{k = 0}^n {( - 1)^k {n \choose k}(x - k)^{n - 1} {\mathop{\rm sgn}} (x - k)},\;\; 0 < x < n. 
$$
Normal approximation (general setting). Suppose that $X_1,X_2,\ldots$ is a sequence of independent and identically distributed random variables with mean $\mu$ and variance $\sigma^2$. Define $S_n = \sum\nolimits_{i = 1}^n {X_i }$. Then, by the central limit theorem,
$$
\frac{{S_n  - n\mu }}{{\sigma \sqrt n }} \to {\rm N}(0,1),
$$
as $n \to \infty$. This means that, for any $x \in \mathbb{R}$,
$$
{\rm P}\bigg(\frac{{S_n  - n\mu }}{{\sigma \sqrt n }} \le x \bigg) \to \Phi (x),
$$ 
where $\Phi$ denotes the distribution function of the ${\rm N}(0,1)$ distribution.
Thus, 
$$
{\rm P}(S_n  \le n\mu  + \sigma \sqrt n x) \to \Phi (x).
$$
Note that the first convergence above gives rise to the approximation
$$
S_n \approx n \mu + \sigma \sqrt{n} Z,
$$
where $Z \sim {\rm N}(0,1)$.
Back to your specific question, you just need to substitute the expectation and variance of a uniform$(0,1)$ random variable. In the context of normal approximation, consider also the four plots given in the MathWorld link above.
