Problem 36 - A Course in Calculus and Real Analysis

Question

Assuming that \(f\) is integrable on \([0,1]\), show that $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) d x . $$

Step-by-Step Solution

Answer

Considering that the expression \(\frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]\) represents a Riemann sum with \(\Delta x = \frac{1}{n}\) and \(x_i^* = \frac{i}{n}\), and knowing that the integral of a function f on the interval [0, 1] is defined as the limit of its Riemann sum when \(n \rightarrow \infty\), we can conclude that: $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) d x $$

Step 1: Understand the problem

We are given a function f and its integral on the interval [0,1]. The goal is to show that the limit of the Riemann sum is equal to the integral of f over the interval [0,1]. The Riemann sum is given as: $$ \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right] $$ We want to show that: $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) d x . $$

Step 2: Recognize the Riemann sum

Recall that the Riemann sum is defined as: $$ R_n = \sum_{i=1}^n f(x_i^*)(\Delta x) $$ Where \(\Delta x = \frac{1}{n}\) and \(x_i^* = \frac{i}{n}\). Compare this definition to what we have in the problem: $$ \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right] $$ We can see a match between the Riemann sum and the expression: the sum goes from \(i=1\) to \(n\), the function f is evaluated at \(x_i^* = \frac{i}{n}\), and the interval \(\Delta x = \frac{1}{n}\).

Step 3: Use the definition of the integral

Now we know that our expression represents a Riemann sum. Recall that the definite integral is defined as the limit of the Riemann sum when \(n \rightarrow \infty\): $$ \int_{0}^1 f(x) \, dx = \lim_{n \rightarrow \infty} R_{n} $$ In our case, we want to show that: $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) d x $$

Step 4: Show the equivalence of the expressions

Now we can use the definition of the integral as the limit of the Riemann sum to show that the two expressions are equal: $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right] = \lim_{n \rightarrow \infty} R_{n} $$ Since the given function f is integrable on [0,1], by the definition of the integral, the limit of the Riemann sum exists and is equal to the integral of f over the interval [0,1]: $$ \lim_{n \rightarrow \infty} R_{n} = \int_{0}^1 f(x) \, dx $$ Combining these two equalities, we have shown the desired result: $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) d x $$

Key Concepts

Integrable Functions

In the realm of calculus, integrable functions play a pivotal role in the comprehension of mathematical analysis. An integrable function is one for which the integral—essentially the area under the curve represented by the function—can be computed over a given interval. To be more specific, for a function to be integrable on an interval, it must satisfy certain properties such as being bounded and having a finite number of discontinuities within that interval.

One of the fundamental features of integrable functions is that they admit the application of the Riemann sum to approximate the area under the curve. This characteristic paves the way for a conceptual bridge to the definite integral, by taking the limit of these Riemann sums as the partition of the interval becomes infinitely fine. Thus, a function being integrable on a closed interval \[a,b\] suggests that the limit of the Riemann sums as the number of subintervals increases without bound does indeed exist and equals the integral of the function over \[a,b\].

Definite Integral Definition

The definite integral of a function provides a powerful tool to analyze and compute the net 'accumulation' of a quantity, which is often visualized as the area under a curve on a graph. Formally defined, the definite integral of a function \(f\) from \(a\) to \(b\) is the limit of Riemann sums as the number of intervals \(n\) approaches infinity, expressed in the iconic notation:

\[\int_{a}^{b} f(x) \, dx\]

This expression captures the cumulative effect of the function \(f\) across the interval \[a,b\], and is only valid for functions that are integrable in this range. The process of determining the definite integral involves partitioning the interval, evaluating the function at each partition point, multiplying by the width of each subinterval, and summing these products — hence the term 'Riemann sum'. The limit of this summation as the subintervals' width tends to zero leads to the precise value of the integral, opening a window into the continuous nature of accumulation.

Limit of a Sequence

At the heart of understanding calculus is the concept of a limit of a sequence. A sequence is an ordered list of numbers, and the limit refers to the value that the terms of the sequence approach as the index increases without bound. Mathematically, if \(a_n\) is a sequence, and \(L\) is a real number, we say that the limit of \(a_n\) as \(n\) approaches infinity is \(L\), and write it as:

\[\lim _{n \rightarrow \infty} a_n = L\]

If such a limit exists, the sequence is said to be convergent; otherwise, it is divergent. This notion is especially crucial when dealing with Riemann sums, where we consider the limit of the sum as the number of partitions approaches infinity. In the context of our original exercise, the sequence in question is the sequence of Riemann sums for finer and finer partitions, converging to the integral, which epitomizes the area under the function \(f\) over the interval \[0,1\]. Understanding sequence limits aids in appreciating the fundamental connection between discrete sums and continuous integrals.