Difference between Sequence and Series

Sequence

An ordered list of numbers that follows a rule is a sequence, and each number in that list is a term. The order carries meaning, because swapping two neighbouring terms changes what the rule produces next.

Terms and Order in a Sequence

The rule decides which numbers appear and how the list continues, so two different rules produce two different sequences, as in the two lists below.

• Sequence of even numbers: $2, 4, 6, 8, \dots$
• Sequence of squares: $1, 4, 9, 16, \dots$

In both examples the sequence records each separate term and the rule that produces the next one.

Series

Adding the terms of a sequence produces a series, and the series is written as a sum expression. The partial sum $S_n$ is the sum of the first $n$ terms, so it changes as $n$ grows. Keeping the two ideas apart is exactly what the comparison below tests.

A Series as a Sum

A series adds the terms of a sequence, and the plus signs between the terms show that addition. The two sums below add the even numbers and the square numbers one term at a time.

• Series of even numbers: $2 + 4 + 6 + 8 + \dots$
• Series of squares: $1 + 4 + 9 + 16 + \dots$

When the terms stop after the $n$-th term, the value of the sum is the partial sum, usually written $S_n$.

Difference between a Sequence and a Series

The table puts the two side by side. A sequence keeps the terms in order, and a series adds those terms into one value.

Partial Sums of Sequence Terms

The sum of the first $n$ terms of a sequence is its $n$th partial sum:

A sequence $(a_n)$ therefore creates a new sequence $(S_n)$ of partial sums. For $a_n=n$:

Adding the terms of $a_n=n$ produces a second sequence, $1, 3, 6, 10, \dots$, and those partial sums grow faster than the original terms.

Finite and Infinite Sums

$S_{20}$ is a finite sum of twenty terms. For an infinite series, we instead ask whether $S_n$ approaches a limit as $n\to\infty$.

An infinite series is defined through the limit of its partial sums. When those partial sums approach one fixed value, the series converges. The topic convergent and divergent series uses this limit to separate series that settle from those that do not.

Exercise

Given the sequence $a_n=2n$. Write its first four terms, then determine its first four partial sums. Keep the two lists apart, because a term describes one position while a partial sum totals every term up to that position.

1. Write its first four terms.
2. Determine its first four partial sums.

Solution

Substituting $n=1,2,3,4$ into $a_n=2n$ gives the first four terms, and adding those terms one by one gives the partial sums.