Calculus How To

Continuous Series & Discrete Series

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The term “continuous series” is used informally, so there isn’t a single definition. Depending on the author, it might mean:

  1. A series with no breaks,
  2. As a synonym for an infinite series,
  3. Where frequencies are given with the variable value (as class intervals).

In sum, you’ll want to read up on the authors intention before deciding on an exact definition.

1. A Continuous Series with No Breaks

A continuous series can be defined as one with has no break or gap in the series. For example, a line segment from two points a to b is the set of all of points between a and b.

line segment

Two endpoints A and B on a line segment, representing a continuous series of points from a to b.



Another example: the class of all points (x, y) in a square (including boundaries), arraned in order of magnitude of the x’s (Huntingdon, 2017).

Using the same logic, a discrete series has definite gaps or breaks in between one point and the next. For example, 1, 3, 5, 7,…19 (Singh, 2008).


When used in this sense, what we’re really talking about here is a string of continuous variables (rather than a “continuous series”).


2. Continuous Series with Class Intervals

In this definition (sometimes found in statistics and economics), data is divided into continuous groups. It’s similar to the definition in #1 above, except that the groups are placed into a frequency distribution table.
continuous series

3. Synonym for Infinite Series

Another definition is simply as another name for “infinite series“. For example, in their article The continuous series of critical points of the two-matrix model at N → ∞ in the double scaling limit, Balaska et al. clearly mean infinity (because of the → ∞) in the title. In their 1974 book, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Keith Oldham and Jerome Spanier mention “…a continuous series of continuously differentiable functions,” which likely falls into the realm of this particular definition.

References

Balaska, S. et al. (1998). The continuous series of critical points of the two-matrix model at N → ∞ in the double scaling limit. Nuclear Physics, Section B, Volume 520, Issue 1, p. 411-432.
Huntingdon, E. (2017). The Continuum and Other Types of Serial Order: Second Edition. Courier Dover Publications.
Oldham, K. & Spanier, J. (1974). The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier Science.
Singh, S. (2008). Biostatistics And Introductory Calculus. Nirali Prakashan.

CITE THIS AS:
Stephanie Glen. "Continuous Series & Discrete Series" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/continuous-series/
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