series
A series is what you get when you add up the terms of a sequence. If a sequence lists the floors of a building, the series is the total height you have climbed once you stack them all together.
A finite series adds finitely many terms and always produces an ordinary number. An infinite series proposes to add infinitely many; that only makes sense as the limit of its running totals (its partial sums), and sometimes that limit does not exist.
Keep the two ideas separate: the sequence is the list, the series is the sum of the list. The sequence 1, 2, 3, 4 has four entries; the series 1 + 2 + 3 + 4 has the single value 10. We often write a series with summation (sigma) notation to keep it compact.
The sequence 2, 4, 6, 8 gives the finite series 2 + 4 + 6 + 8 = 20.
Adding the terms of a sequence turns a list into a single sum.
“Series” is one of those English words that is the same in singular and plural — one series, two series. In Chinese it is 级数 / 級數 in either case.