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 is a non-zero real number, and represents the greatest integer less than or equal to .
Step 1: Analyze the summation
The summation in the numerator is:
[ S_n = [r] + [2r] + [3r] + \dots + [nr]. ]
The term refers to the greatest integer less than or equal to , and for large , we need to understand how this sum behaves relative to .
Step 2: Break down the sum
For large , the value can be written as:
where is the fractional part of , which lies in the interval .
Thus, the sum becomes:
The first part of the sum is:
The second part is a sum of fractional parts, and since , the sum grows linearly with , i.e., it is approximately .
Step 3: Simplify the limit
Thus, for large , we have:
Dividing by , we get:
As , the term vanishes, and we are left with:
Final Answer:
Do you want further details or have any questions?
Here are 5 related questions to consider:
- What happens if is an integer in the given limit problem?
- How does the behavior of the fractional part affect the limit?
- Can this technique be applied to other sums involving greatest integer functions?
- What if we replace with , where is a different power?
- 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
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