It is critical to comprehend the difference between a sequence and a series.
In essence, a function which maps each positive integer (sometimes including zero) to each term in an ordered list of values. The domain of a sequence is the set of all positive integers.
For instance, the sequence $\{a_n\} = 2, 4, 6, 8, 10, ..., a_n$ maps each positive integer to the rule $2n$.
Thus, the terms of the sequence are: $$ \{a_n\} = \\ a_1 \text{ -> } 2,\\ a_2 \text{ -> } 4,\\ a_3 \text{ -> } 6,\\ a_4 \text{ -> } 8,\\ ... $$
The sum of an infinite sequence of numbers, e.g.:
$$ S_n = \sum_{n=1}^{\infty} a_n $$A sequence and series pairing which you will use often is an infinite series & the sequence of its partial sums. The first is the infinite series itself, $\{a_n\}$, and the second it the sequence of the series' partial sums:
$$ \sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + ... + a_n \\ \{s_n\} = \\ s_1 = a_1, \\ s_2 = a_1 + a2, \\ s_3 = a_1 + a_2 + a_3, \\ s_4 = a_1 + a_2 + a_3 + a_4, \\ s_5 = a_1 + a_2 + a_3 + a_4 + a_5, \\ ..., \\ s_n = a_1 + a_2 + a_3 + a_4 + a_5 + ... + a_n $$
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