A function that is either a sine or cosine, i.e. we don't know or care where the origin is exactly.

This is particularly relevant in electronics, where the oscilloscope's time origin is set to match the wave.

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A series adds the terms of a sequence in their specified order. Its infinite-sum notation is defined through finite partial sums:

S_{N} = \sum_{n = 1}^{N} a_{n}, \sum_{n = 1}^{\infty} a_{n} = lim_{N \to \infty} S_{N}

when the displayed limit of a sequence exists as a finite number. Otherwise the series is not a convergent series in the ordinary sense. Writing an infinite list of additions does not by itself assign it a finite sum. Cesàro summation is a different prescription, with its own definition.

Two positive series make the distinction concrete. For

a_{n} = 1/ n

, the individual terms tend to zero but the harmonic series has unbounded partial sums. For

a_{n} = 1/ [n (n + 1)]

, partial fraction decomposition reveals a telescoping series whose partial sums tend to one. The useful question is how much all the remaining terms can contribute together, rather than only how small the next term looks.

In mathematics, a series is the sum of the terms of a sequence. A sequence is an ordered list of numbers, and when you sum these numbers together, you form a series. There are different types of series, including: 1. **Finite Series**: This involves summing a finite number of terms.