Key Concepts and Formulas

• Greatest Integer Function Property: For any real number $x$, the greatest integer function $[x]$ satisfies the inequality $x - 1 \le [x] < x$.
• Sum of an Arithmetic Series: The sum of the first $n$ terms of an arithmetic series $a, a+d, a+2d, \dots$ is given by $S_n = \frac{n}{2}(2a + (n-1)d)$. For the series $1, 2, \dots, n$, the sum is $\frac{n(n+1)}{2}$.
• Squeeze Theorem (Sandwich Theorem): If $f(x) \le g(x) \le h(x)$ for all $x$ in some open interval containing $c$, except possibly at $c$ itself, and if $\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} h(x) = L$, then $\mathop {\lim }\limits_{x \to c} g(x) = L$.

Step-by-Step Solution

1. Apply the Greatest Integer Function Property: We are given the expression $\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$. For each term $[kr]$ in the sum, where $k$ is an integer from 1 to $n$, we can apply the property $x - 1 \le [x] < x$. Thus, for each $k \in \{1, 2, \dots, n\}$, we have: $kr - 1 \le [kr] < kr$

2. Sum the Inequalities: We will sum these inequalities for $k$ from 1 to $n$ to get bounds for the numerator. Adding $kr - 1 \le [kr]$ for $k=1, \dots, n$: $\sum_{k=1}^{n} (kr - 1) \le \sum_{k=1}^{n} [kr]$ $r \sum_{k=1}^{n} k - \sum_{k=1}^{n} 1 \le \sum_{k=1}^{n} [kr]$ Using the sum of the first $n$ integers formula, $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$, and $\sum_{k=1}^{n} 1 = n$: $r \frac{n(n+1)}{2} - n \le \sum_{k=1}^{n} [kr]$

   Adding $[kr] < kr$ for $k=1, \dots, n$: $\sum_{k=1}^{n} [kr] < \sum_{k=1}^{n} kr$ $\sum_{k=1}^{n} [kr] < r \sum_{k=1}^{n} k$ $\sum_{k=1}^{n} [kr] < r \frac{n(n+1)}{2}$

   Combining these, we get the bounds for the sum in the numerator: $r \frac{n(n+1)}{2} - n \le [r] + [2r] + ... + [nr] < r \frac{n(n+1)}{2}$

3. Divide by $n^2$ to form the expression in the limit: Now, we divide all parts of the inequality by $n^2$ to match the form of the limit expression. $\frac{r \frac{n(n+1)}{2} - n}{n^2} \le \frac{[r] + [2r] + ... + [nr]}{n^2} < \frac{r \frac{n(n+1)}{2}}{n^2}$

4. Evaluate the Limits of the Lower and Upper Bounds: We will now evaluate the limits of the lower and upper bound expressions as $n \to \infty$.

   For the lower bound: $\mathop {\lim }\limits_{n \to \infty } \frac{r \frac{n(n+1)}{2} - n}{n^2} = \mathop {\lim }\limits_{n \to \infty } \frac{\frac{r}{2}(n^2+n) - n}{n^2}$ $= \mathop {\lim }\limits_{n \to \infty } \frac{\frac{r}{2}n^2 + \frac{r}{2}n - n}{n^2}$ Divide each term by $n^2$: $= \mathop {\lim }\limits_{n \to \infty } \left( \frac{\frac{r}{2}n^2}{n^2} + \frac{(\frac{r}{2}-1)n}{n^2} \right)$ $= \mathop {\lim }\limits_{n \to \infty } \left( \frac{r}{2} + \frac{\frac{r}{2}-1}{n} \right)$ As $n \to \infty$, $\frac{\frac{r}{2}-1}{n} \to 0$. $= \frac{r}{2} + 0 = \frac{r}{2}$

   For the upper bound: $\mathop {\lim }\limits_{n \to \infty } \frac{r \frac{n(n+1)}{2}}{n^2} = \mathop {\lim }\limits_{n \to \infty } \frac{\frac{r}{2}(n^2+n)}{n^2}$ $= \mathop {\lim }\limits_{n \to \infty } \left( \frac{\frac{r}{2}n^2}{n^2} + \frac{\frac{r}{2}n}{n^2} \right)$ $= \mathop {\lim }\limits_{n \to \infty } \left( \frac{r}{2} + \frac{r}{2n} \right)$ As $n \to \infty$, $\frac{r}{2n} \to 0$. $= \frac{r}{2} + 0 = \frac{r}{2}$

5. Apply the Squeeze Theorem: We have established the following inequality: $\frac{r \frac{n(n+1)}{2} - n}{n^2} \le \frac{[r] + [2r] + ... + [nr]}{n^2} < \frac{r \frac{n(n+1)}{2}}{n^2}$ And we found that the limit of the lower bound is $\frac{r}{2}$ and the limit of the upper bound is $\frac{r}{2}$. Therefore, by the Squeeze Theorem, the limit of the middle expression must also be $\frac{r}{2}$. $L = \mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}} = \frac{r}{2}$

Common Mistakes & Tips

• Incorrectly Applying the Greatest Integer Inequality: Ensure you use the correct form $x-1 \le [x] < x$. Using $[x] \le x$ or $[x] > x-1$ might lead to incorrect bounds.
• Algebraic Errors in Summation or Limit Evaluation: Carefully expand and simplify terms, especially when dealing with polynomials in $n$. Pay close attention to the highest power of $n$ when evaluating limits of rational functions.
• Forgetting to Divide by $n^2$: The problem requires the limit of the expression divided by $n^2$. Make sure to divide all parts of the inequality by $n^2$ before taking the limit.

Summary

The problem asks for the limit of a sum involving the greatest integer function. We utilized the property of the greatest integer function ($x-1 \le [x] < x$) to establish lower and upper bounds for the sum $[r] + [2r] + \dots + [nr]$. After summing these inequalities and dividing by $n^2$, we applied the Squeeze Theorem. By evaluating the limits of the lower and upper bounds as $n \to \infty$, we found that both limits converge to $\frac{r}{2}$. Consequently, the limit of the original expression is also $\frac{r}{2}$.

The final answer is $\boxed{\frac{r}{2}}$.