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If u₁, u₂, …, u_i, … constitute a given (infinitely long) sequence of numbers, one often needs to calculate the sum of the first n terms thereof – or nth partial sum, S_n, defined as

(2.72)

and also known as series; if the partial sums S₁, S₂, …, S_i, … converge to a finite limit, say, S, according to

(2.73)

then S can be viewed as the infinite series

(2.74)

– while the said series is termed convergent. Should the sequence of partial sums tend to infinite, or oscillate either finitely or infinitely, then the series would be termed divergent.

Despite the great many series that may be devised, two of them possess major practical importance – arithmetic and geometric progressions, as well as their hybrid (i.e. arithmetic–geometric progressions); hence, all three types will be treated below in detail.