Problem 79 - Essential Calculus

Question

In your own words, define each of the following. (a) Sequence (b) Convergence of a sequence (c) Monotonic sequence (d) Bounded sequence

Step-by-Step Solution

Answer

A sequence is an ordered collection of objects. A sequence is said to converge if it approaches a specific value as the number of terms goes to infinity. A sequence is called monotonic if it always either increases or decreases, and it is called bounded if all its terms lie within a specific range.

Step 1: Define Sequence

A sequence in mathematics is an enumerated collection of objects in a specific order. The objects in the sequence are often called terms, and the term in each individual position is identified by a positive integer, often representing the position of the term in the sequence.

Step 2: Define Convergence of a Sequence

The convergence of a sequence is a property that a sequence has when it approaches a specific value as the number of terms goes to infinity. In other words, you can say a sequence converges to a value 'L' if the terms of the sequence become arbitrarily close to 'L' as you go further and further into the sequence.

Step 3: Define Monotonic Sequence

A monotonic sequence is one that either entirely increases or entirely decreases. A sequence is called monotonically increasing if each term is greater than or equal to the previous one, and it's called monotonically decreasing if each term is less than or equal to the previous one.

Step 4: Define Bounded Sequence

A sequence is known as a bounded sequence if there is a real number M such that every term in the sequence is less than or equal to M. Essentially, it means that all the terms of the sequence lie within a specific range.