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A definite integral is a fundamental concept in calculus that represents the area under a curve, described by a function, within a specific interval. This area is calculated by integrating the function over the interval using limits of integration. In mathematical terms, for a function \( f(x) \) defined on an interval \([a, b]\), the definite integral is denoted as \( \int_{a}^{b} f(x) \, dx \).

This integral essentially sums up an infinite number of infinitesimally small areas beneath the curve of \( f(x) \) from \( x = a \) to \( x = b \). This is especially useful to compute physical quantities like distance, area, and total accumulated change.

In the context of a Riemann sum, as you increase the number of partitions, \( n \), the approximated sum converges to the actual value of the definite integral. For example, in the exercise, the sequence \( a_n \) is interpreted as a Riemann sum that approximates the definite integral \( \int_{0}^{1} \frac{1}{1+x} \, dx \). As the number of partitions grows infinitely large, this sum becomes an exact representation, thus demonstrating the concept of convergence to the definite integral.

Convergence is a concept in mathematics where a sequence or series approaches a specific value as the number of terms increases. In terms of the exercise, we are examining the convergence of a sequence of Riemann sums \( \{a_n\} \) as \( n \) approaches infinity.

The sequence \( a_n \) is a sum built from dividing the interval \([0, 1]\) into smaller sub-intervals. Each partition gets smaller as \( n \) increases, leading to more precise approximations of a definite integral.

When \( n \to \infty \), the Riemann sum effectively becomes the exact value of the integral \( \int_{0}^{1} \frac{1}{1+x} \, dx \).

• This reflects a key property of integrals: they can be approximated by Riemann sums, which, when summed over an increasingly subdivided interval, converge to the integral's exact value.
• In the given task, it helps establish that \( \lim_{n \to \infty} a_n = \ln 2 \).

An antiderivative, also known as an indefinite integral, is a function that reverses the process of differentiation. In other words, the derivative of an antiderivative returns the original function. For a function \( f(x) \), its antiderivative is denoted by \( F(x) \), such that \( F'(x) = f(x) \).

Finding the antiderivative is crucial for evaluating definite integrals, as it helps in determining the area under a curve. In the exercise provided, identifying the antiderivative of \( \frac{1}{1+x} \) was necessary to evaluate the integral \( \int_{0}^{1} \frac{1}{1+x} \, dx \).

• The antiderivative of \( \frac{1}{1+x} \) is \( \ln(1+x) \).
• To compute the definite integral, substitute the limits of integration: Evaluate \( F(x) \) from 0 to 1, resulting in \( \ln 2 \).

This demonstrates how an antiderivative directly facilitates computing the values of definite integrals, thus playing a pivotal role in understanding the entire integration process.