Problem 56 - Essential Calculus

Question

Define the terms converges and diverges when working with improper integrals.

Step-by-Step Solution

Answer

In the context of improper integrals, the term 'converges' refers to a situation where the values approach a finite number as the variable approaches infinity. On the other hand, 'diverges' refers to situations where the values don't tend towards any specific finite number as the variable approaches infinity.

Step 1: Defining 'Converges'

In the context of improper integrals, a series or function is said to 'converge' if the values approach a finite number as the variable approaches infinity (positive or negative). Mathematically, this can be represented as \(\lim_{{x \to \infty}} f(x) = L\), where L is a finite number. This means that as x increases or decreases without bound, the function f(x) approaches but never reaches the value L.

Step 2: Defining 'Diverges'

A series or function 'diverges' when its values don't tend towards a specific finite value as the variable approaches infinity. This usually happens in two scenarios. Firstly, when as x approaches infinity, f(x) also approaches infinity or negative infinity, represented as \(\lim_{{x \to \infty}} f(x) = \infty\) or \(\lim_{{x \to \infty}} f(x) = -\infty\). Secondly, when the limit as x approaches infinity does not exist, meaning the function doesn't tend towards any specific number (finite or infinite).

Step 3: Applying to Improper Integrals

The concepts of convergence and divergence apply to improper integrals in a similar way. An improper integral converges when the definite integral from a to infinity, \(\int_a^{\infty} f(x) dx\), or from negative infinity to a, \(\int_{-\infty}^a f(x) dx\), exists and equals a finite number. If the integral does not exist or equals infinity, then the improper integral diverges.