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A sequence is an ordered list of numbers or terms, arranged in a specific order. Each number in the sequence is called a term, and it follows a particular rule or pattern. For example, in the sequence 2, 4, 6, 8, ..., the rule is to add 2 to the previous term to get the next term. The key aspect of sequences is the order and arrangement of the terms.

Sequences can be either finite or infinite. A finite sequence has a specific number of terms, while an infinite sequence goes on indefinitely. Understanding the rule behind a sequence helps in identifying and predicting its terms.

A series is the result of summing the terms of a sequence. In other words, if you take the terms of a sequence and add them up, you get a series. For example, if we take the sequence 2, 4, 6, 8, ..., and add these terms, we get the series 2 + 4 + 6 + 8 + ... .

Like sequences, series can also be finite or infinite. A finite series has a definite number of summed terms, while an infinite series continues sum indefinitely.

Series are essential in various fields of mathematics, including calculus and number theory, and they often help understand the behavior and properties of sequences more deeply.

The sum of sequence terms is what forms a series. By adding each term in the sequence, you derive the series. This sum can be represented mathematically using the notation \(\sum\), known as the summation symbol. For instance, the sum of the first four terms of the sequence 2, 4, 6, 8 is \[ 2 + 4 + 6 + 8 = 20 \].

To sum an infinite sequence's terms, we often look at the concept of convergence. An infinite series converges if the sum of its terms approaches a specific value as more terms get added. Otherwise, it diverges.

Various formulas and methods are used to assess and compute the sum of series, especially when dealing with arithmetic or geometric sequences. Knowing how to calculate these sums is crucial for deeper mathematical problems and applications.