The value of lim n→∞ ([r]+[2r]+.....+[nr]) / n², where r is non-zero real number and [r] denotes the

Q: Wait, why can't I just approximate $[kr] \approx kr$ and solve it? Why do I need the inequality?

That is a classic trap! In limits, especially when $n$ approaches infinity, small errors can accumulate. If you just replace $[kr]$ with $kr$, you are ignoring the fractional part, which can be up to $1$. When you sum $n$ such terms, that error becomes $n$, and when you divide by $n^2$, the error term $n/n^2$ vanishes, but you must prove it rigorously. The Sandwich Theorem is the only way to mathematically guarantee that the 'jaggedness' of the GIF doesn't affect the final limit.

Q: Does the Sandwich Theorem always work for these types of sums?

It is the gold standard for sums involving the Greatest Integer Function. Whenever you see a sum of GIF terms where the argument is linear in $k$, the Sandwich Theorem is your best friend. It allows you to bound the discrete sum between two continuous, easily integrable (or summable) functions.

Q: What if $r$ is negative? Does the inequality $x-1 < [x] \le x$ still hold?

Yes, absolutely! The inequality $x-1 < [x] \le x$ is a universal property of the Greatest Integer Function, regardless of whether $x$ is positive, negative, or zero. The logic holds perfectly. The beauty of this method is its robustness; it doesn't care about the sign of $r$.

Question

The value of $\lim_{n 	o \infty} \frac{[r]+[2r]+\dots+[nr]}{n^2}$, where $r$ is non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r$, is equal to :

Options:

(A) $\frac{r}{2}$

(B) $r$

(C) $2r$

(D) $0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The problem asks us to evaluate the limit:
$$\lim_{n 	o \infty} \frac{[r] + [2r] + \dots + [nr]}{n^2}$$

The Greatest Integer Function (GIF), denoted by $[x]$, is inherently discontinuous and difficult to manipulate directly. To overcome this, we employ the **Sandwich Theorem** (or Squeeze Theorem).

### The Strategy of the Sandwich

We utilize the fundamental property of the GIF, which states that for any real number $x$:
$$x - 1 < [x] \le x$$

This inequality allows us to "cage" the jagged function between two smooth, continuous expressions. By applying this to every term in the series, we transform a discrete, intractable problem into a manageable one.

### The Summation Battle

We apply the inequality to each term $kr$ for $k = 1, 2, \dots, n$:
$$kr - 1 < [kr] \le kr$$

Summing these inequalities vertically from $k=1$ to $n$, we obtain:
$$\sum_{k=1}^{n} (kr - 1) < \sum_{k=1}^{n} [kr] \le \sum_{k=1}^{n} kr$$

Using the standard summation formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$, the right side becomes:
$$\sum_{k=1}^{n} kr = r \frac{n(n+1)}{2}$$

The left side is simply the sum of $kr$ minus the sum of $1$ repeated $n$ times:
$$\sum_{k=1}^{n} (kr - 1) = r \frac{n(n+1)}{2} - n$$

### The Squeeze

We now have the following inequality for our target expression:
$$r \frac{n(n+1)}{2} - n < \sum_{k=1}^{n} [kr] \le r \frac{n(n+1)}{2}$$

To find the limit, we divide the entire inequality by $n^2$:
$$\frac{r n(n+1)}{2n^2} - \frac{n}{n^2} < \frac{\sum_{k=1}^{n} [kr]}{n^2} \le \frac{r n(n+1)}{2n^2}$$

Simplifying the bounds, we get:
$$\frac{r}{2} \left( 1 + \frac{1}{n} ight) - \frac{1}{n} < \frac{\sum_{k=1}^{n} [kr]}{n^2} \le \frac{r}{2} \left( 1 + \frac{1}{n} ight)$$

### The Final Victory

As $n 	o \infty$, the term $\frac{1}{n}$ approaches $0$. Consequently, both the lower bound and the upper bound converge to the same value:
$$\lim_{n 	o \infty} \left[ \frac{r}{2} \left( 1 + \frac{1}{n} ight) - \frac{1}{n} ight] = \frac{r}{2}$$
$$\lim_{n 	o \infty} \left[ \frac{r}{2} \left( 1 + \frac{1}{n} ight) ight] = \frac{r}{2}$$

By the **Sandwich Theorem**, the limit of the target expression is forced to the same value. Thus, the final result is:
$$\mathbf{\frac{r}{2}}$$

Animated Solution for Mathematics - Limits, Continuity and Differentiability: The value of $lim_{n \to \infty} \frac{[ r ] + [ 2 r ] + \dots + [ n r ]}{n ^{2}}$ , where $r$ is non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r$ , is equal to :

Select Answer:

Visualized Solution

Understanding the Limit

The G.I.F. Inequality

Applying to the Series Terms

Summing the Inequalities

Simplifying the Upper Bound

Simplifying the Lower Bound

Constructing the Sandwich

Simplifying the Fractions

Limit of the Upper Bound

Limit of the Lower Bound

Applying Sandwich Theorem

The Sigma Insight: Evaluation of Limits & L'Hopital's Rule

Analyzing the Setup

The Strategy of the Sandwich

The Summation Battle

The Squeeze

The Final Victory

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