We are given the following limit expression:

$\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{1 + \left(\frac{i}{n}\right)^2}$

This expression resembles the Riemann sum for a definite integral. Let's explore this step by step:

Step 1: Identify the form of the Riemann sum

A Riemann sum for the integral of a function $f(x)$ over an interval $[a, b]$ is generally of the form:

$\sum_{i=1}^{n} f(x_i) \Delta x$

where $x_i$ is a sample point in the subinterval and $\Delta x = \frac{b - a}{n}$ is the width of each subinterval.

Step 2: Compare with the given sum

In our expression:

• $\frac{1}{n}$ corresponds to $\Delta x$, the width of each subinterval.
• The sum involves the function $\frac{1}{1 + \left(\frac{i}{n}\right)^2}$, where the term $\frac{i}{n}$ represents the sample point $x_i$.
• The range of the sum is from $i = 1$ to $i = n$, which suggests that the sample points $\frac{i}{n}$ are partitioning the interval $[0, 1]$.

Thus, we can interpret the sum as a Riemann sum for the integral of the function $\frac{1}{1 + x^2}$ over the interval $[0, 1]$.

Step 3: Express the limit as a definite integral

As $n \to \infty$, the Riemann sum converges to the following definite integral:

$\int_0^1 \frac{1}{1 + x^2} \, dx$

Final Answer:

The limit expression can be written as the following definite integral:

$\int_0^1 \frac{1}{1 + x^2} \, dx$

This integral is a standard integral, and its value can be found using basic calculus.

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Would you like to compute the value of this integral, or do you have any questions about the steps involved?

Here are some related questions for further exploration:

1. What is the result of the integral $\int_0^1 \frac{1}{1 + x^2} \, dx$?
2. How can we evaluate Riemann sums for different functions and intervals?
3. What is the geometric interpretation of the function $\frac{1}{1 + x^2}$ in the context of integration?
4. How can we relate Riemann sums to numerical integration techniques such as the trapezoidal rule?
5. Can we derive the general form of a Riemann sum for an arbitrary function over any interval?

Tip: When working with Riemann sums, recognize patterns that suggest a definite integral by comparing the sum to the general form $\sum f(x_i) \Delta x$.