What is the difference between a sequence and a series? This is a common question in mathematics, especially when it comes to understanding the concepts of convergence and divergence. While both sequences and series are fundamental components of calculus and analysis, they represent different mathematical structures and have distinct properties.
A sequence is an ordered list of numbers, where each number is called a term. The terms in a sequence are indexed by natural numbers, starting from 1. For example, the sequence (1, 2, 3, 4, 5, …) consists of the first five natural numbers. Sequences can be finite or infinite, and they can be arithmetic, geometric, or any other type of progression.
On the other hand, a series is the sum of the terms of a sequence. In other words, a series is the result of adding up all the terms in a sequence. For instance, the series 1 + 2 + 3 + 4 + 5 + … is the sum of the first five natural numbers. Like sequences, series can be finite or infinite, and they can be arithmetic, geometric, or other types.
One key difference between sequences and series is that a sequence is a list of numbers, while a series is a single number obtained by adding the terms of the sequence. This distinction is important because it affects how we analyze and evaluate sequences and series.
Another significant difference lies in their convergence properties. A sequence is said to converge if its terms approach a specific limit as the index approaches infinity. In contrast, a series is considered convergent if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. It’s important to note that a convergent series is always associated with a convergent sequence, but the reverse is not necessarily true.
For example, consider the sequence 1/n, where n is a natural number. This sequence converges to the limit 0 as n approaches infinity. However, the series 1 + 1/2 + 1/3 + 1/4 + … (also known as the harmonic series) is divergent, meaning its sum does not approach a finite limit.
In conclusion, while sequences and series are closely related, they are distinct mathematical concepts. A sequence is an ordered list of numbers, while a series is the sum of the terms in a sequence. Convergence properties also differ between the two, with a convergent series always associated with a convergent sequence but not vice versa. Understanding these differences is crucial for a deeper comprehension of calculus and analysis.
