What is the difference between a sequence and a series

Sequence vs. Series

A sequence is a function whose domain is the natural numbers, assigning a single term to each index. It is written ($a_1$, $a_2$, $a_3$,\dots) or \{$a_n$\}_{n=1}^{$\infty$}. For example, the sequence defined by $a_n$ = 2n+1 yields 3,5,7,9,\dots. The focus is on the individual terms and their behavior as n grows. One often studies whether the terms approach a limit; if \lim_{n\to\infty} $a_n$ = L, the sequence converges to L. If no such limit exists, the sequence diverges. Sequences can be arithmetic, geometric, or defined recursively, each with its own formula for the nth term. Understanding a sequence’s pattern is the first step before forming a series from it. In calculus, sequences provide the foundation for limits, continuity, and series expansions.

Series as a Sum of a Sequence

A series takes a sequence \{$a_n$\} and adds its terms together, forming the expression \sum_{n=1}^{$\infty$} $a_n$. The sum is not computed directly; instead one examines the sequence of partial sums $S_N$ = \sum_{n=1}^{N} $a_n$. If the partial sums approach a finite limit S as N\to\infty, the series converges to S; otherwise it diverges. For instance, the geometric series with $a_n$ = $\frac{1}{2^{n}}$ has partial sums $S_N$ = 1 - $\frac{1}{2^{N}}$ and converges to 1. Series are central to power series, Fourier series, and Taylor expansions, where each term contributes to an approximation of a function. Because a series collapses many terms into a single value, its convergence properties are more delicate than those of the underlying sequence.

Key differences between a sequence and a series:

• A sequence lists individual terms; a series adds them together
• Sequences are indexed by n; series are expressed with the sigma notation
• Convergence of a sequence looks at \lim_{n\to\infty} $a_n$; convergence of a series looks at \lim_{N\to\infty} $S_N$
• A divergent sequence can still generate a convergent series if terms decrease fast enough

Testing convergence of a series:

1. 1Write the series in sigma notation and identify $a_n$
2. 2Form the partial sum $S_N$ = \sum_{n=1}^{N} $a_n$
3. 3Take the limit \lim_{N\to\infty} $S_N$; if it exists and is finite, the series converges
4. 4If the limit does not exist or is infinite, the series diverges

Comparison of properties:

Example: consider the sequence $a_n$ = 2n+1. The first five terms are 3, 5, 7, 9, 11. The corresponding series \sum_{n=1}^{5} $a_n$ adds these numbers: 3+5+7+9+11 = 35. The partial sum formula for this arithmetic sequence is $S_N$ = $\frac{N}{2}$($a_1$ + $a_N$). Plugging N=5, $a_1$=3, $a_5$=11 gives $S_5$ = $\frac{5}{2}$(3+11)=$\frac{5}{2}$\times14=35, confirming the direct addition. If we let N\to\infty, the sum grows without bound, so the infinite series diverges.