Math Problem Statement

The value of \mathop {\lim }\limits_{n \to \infty } \frac{{[r] + [2r] + ... + [nr]}}{{{n^2}}}, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

0

r

\frac{r}{2}

2r

Solution

We are tasked with finding the value of:

[ \lim_{n \to \infty } \frac{[r] + [2r] + \dots + [nr]}{n^2}, ] where rr is a non-zero real number, and [x][x] represents the greatest integer less than or equal to xx.

Step 1: Analyze the summation

The summation in the numerator is:

[ S_n = [r] + [2r] + [3r] + \dots + [nr]. ]

The term [kr][kr] refers to the greatest integer less than or equal to krkr, and for large nn, we need to understand how this sum behaves relative to n2n^2.

Step 2: Break down the sum

For large kk, the value [kr][kr] can be written as:

[kr]=kr{kr},[kr] = kr - \{kr\},

where {kr}\{kr\} is the fractional part of krkr, which lies in the interval [0,1)[0, 1).

Thus, the sum becomes:

Sn=(r+2r+3r++nr)({r}+{2r}+{3r}++{nr}).S_n = (r + 2r + 3r + \dots + nr) - (\{r\} + \{2r\} + \{3r\} + \dots + \{nr\}).

The first part of the sum is:

r(1+2+3++n)=rn(n+1)2rn22 for large n.r(1 + 2 + 3 + \dots + n) = r \cdot \frac{n(n+1)}{2} \sim \frac{r n^2}{2} \text{ for large } n.

The second part is a sum of fractional parts, and since {kr}[0,1)\{kr\} \in [0, 1), the sum {r}+{2r}++{nr}\{r\} + \{2r\} + \dots + \{nr\} grows linearly with nn, i.e., it is approximately O(n)O(n).

Step 3: Simplify the limit

Thus, for large nn, we have:

Snrn22O(n).S_n \sim \frac{r n^2}{2} - O(n).

Dividing by n2n^2, we get:

Snn2r2O(n)n2.\frac{S_n}{n^2} \sim \frac{r}{2} - \frac{O(n)}{n^2}.

As nn \to \infty, the term O(n)n2\frac{O(n)}{n^2} vanishes, and we are left with:

limnSnn2=r2.\lim_{n \to \infty} \frac{S_n}{n^2} = \frac{r}{2}.

Final Answer:

r2.\boxed{\frac{r}{2}}.

Do you want further details or have any questions?

Here are 5 related questions to consider:

  1. What happens if rr is an integer in the given limit problem?
  2. How does the behavior of the fractional part affect the limit?
  3. Can this technique be applied to other sums involving greatest integer functions?
  4. What if we replace n2n^2 with npn^p, where pp is a different power?
  5. How does this limit relate to the concept of Riemann sums in integral calculus?

Tip: For large sums involving step functions, consider separating the integer and fractional parts to simplify the analysis.

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Math Problem Analysis

Mathematical Concepts

Limits
Summation
Greatest Integer Function
Asymptotic Behavior

Formulas

\lim_{n \to \infty} \frac{[r] + [2r] + ... + [nr]}{n^2}
S_n = r(1 + 2 + ... + n) - \{r\} - \{2r\} - ... - \{nr\}
n(n+1)/2 ≈ n^2/2 for large n

Theorems

Asymptotic Analysis
Summation of Arithmetic Series

Suitable Grade Level

Undergraduate Mathematics