lim (n→∞) 3/n Σ_(r=0)^(n-1) (2 + r/n)² is equal to (1) 12, (2) 19/3, (3) 0, (4) 19

Q: Wait, why is the integral from $0$ to $1$ and not something else?

Great question! The limits of integration are determined by the behavior of the variable $\frac{r}{n}$ as $n \to \infty$. When $r=0$, $\frac{r}{n} = 0$. When $r=n-1$, $\frac{r}{n} = \frac{n-1}{n} = 1 - \frac{1}{n}$, which approaches $1$ as $n \to \infty$. Thus, the interval is $[0, 1]$.

Q: What happens to the $3$ in the $\frac{3}{n}$ term?

The $3$ is just a constant multiplier. In the limit of a sum, constants can be pulled outside the summation and the integral. We rewrite $\frac{3}{n}$ as $3 \cdot \frac{1}{n}$, where $\frac{1}{n}$ becomes the differential $dx$ and the $3$ remains as a coefficient for the final integral.

Q: Why can't we just use energy conservation or some other physics principle?

This is a pure mathematical limit problem, not a physics problem. While it looks like a physics summation, it is a Riemann sum. We must use the tools of calculus—specifically, the definition of the definite integral—to evaluate it.

Question

$\lim_{n 	o \infty} \frac{3}{n} \sum_{r=0}^{n-1} \left( 2 + \frac{r}{n} ight)^2$ is equal to

Options:

(A) $12$

(B) $\frac{19}{3}$

(C) $0$

(D) $19$

Sigma Percentile · Accepted Answer

### The Beauty of the Infinite Sum

Imagine you are standing on the edge of a vast, infinite landscape, looking at a series of tiny, jagged rectangles. This is the visual reality behind the expression:

$$\lim_{n 	o \infty} \frac{3}{n} \sum_{r=0}^{n-1} \left( 2 + \frac{r}{n} ight)^2$$

It is not just a collection of symbols; it is a story of accumulation. We are summing the areas of $n$ rectangles, and as $n$ grows to infinity, these rectangles smooth out to form a perfect, continuous area under a curve.

This is the fundamental bridge between discrete summation and continuous integration.

### The Bridge to Calculus

To solve this, we invoke the power of the Riemann sum. The general theorem states that:

$$\lim_{n 	o \infty} \frac{1}{n} \sum_{r=0}^{n-1} f\left(\frac{r}{n}ight) = \int_{0}^{1} f(x) \, dx$$

Our goal is to map our expression to this standard form. We identify the continuous variable $x = \frac{r}{n}$, which represents the position along the horizontal axis.

The width of each rectangle, $\frac{1}{n}$, transforms into the infinitesimal differential $dx$ as $n 	o \infty$.

### The Integration Process

We have a constant factor of $3$ outside the summation, which we can safely pull out of the integral. Our expression transforms into:

$$3 \int_{0}^{1} (2+x)^2 \, dx$$

Now, we face the integral of $(2+x)^2$. While we could expand the square, the extended power rule for linear functions is much more elegant.

The integral of $(ax+b)^n$ is $\frac{(ax+b)^{n+1}}{a(n+1)}$. Here, $a=1$, so the integral of $(2+x)^2$ is $\frac{(2+x)^3}{3}$.

Multiplying by our constant $3$, we get:

$$3 \cdot \frac{(2+x)^3}{3} = (2+x)^3$$

### The Final Evaluation

We evaluate this from $0$ to $1$ using the Fundamental Theorem of Calculus.

At the upper limit $x=1$, we have $(2+1)^3 = 3^3 = 27$. At the lower limit $x=0$, we have $(2+0)^3 = 2^3 = 8$.

Subtracting the lower limit value from the upper limit value, we get $27 - 8 = 19$.

The complexity of the infinite sum dissolves into the simplicity of the number **19**. This is the elegance of calculus—taking a chaotic, infinite process and finding the precise, finite truth hidden within.

Animated Solution for Mathematics - Definite Integration: $lim_{n \to \infty} \frac{3}{n} \sum_{r = 0}^{n - 1} (2 + \frac{r}{n})^{2}$ is equal to

Select Answer:

Visualized Solution

Analyze the Summation Structure

Definite Integral as Limit of a Sum

Map the Continuous Variable $x$

Map the Differential $d x$

Adjust the Constant Factor

Determine Lower Limit

Determine Upper Limit

Set up the Definite Integral

Apply the Power Rule

Execute the Integration

Evaluate at Upper Limit

Evaluate at Lower Limit

Final Subtraction

The Sigma Insight: Definite Integral as a Limit of a Sum

The Beauty of the Infinite Sum

The Bridge to Calculus

The Integration Process

The Final Evaluation

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Animated Solution for Mathematics - Definite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limn→∞​n3​∑r=0n−1​(2+nr​)2 is equal to