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In mathematics, we have series, like

1, 2, 3, 4, 5, ...

Should I say 'the series is convergent' or 'the series are convergent'?

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The word 'series' is both a singular and plural form. When it has the singular sense of 'one set', it takes a singular verb. When it has the plural sense of two or more sets, it takes a plural verb. For examples:

A series of lectures is scheduled.

Two series of lectures are scheduled: one for experts and one for laypeople. (The Free Dictionary).

1,2,3,4,5,....

Here we are talking about a set of numbers, so the right sentence is:

The series is convergent.

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1,2,3,4,5....

One series: The series is _____

2,4,6,8,10...

1,4,9,16,25....

Two or more series: The series are _____

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