Calculus How To

Finite Series

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Sequence and Series >

A finite series is a sum of a set amount of terms; A series of numbers (e.g. 1 + 2 + 3) is obtained from a sequence of numbers (e.g. 1, 2, 3); Finite series always have a first term and a last term. Plus, you can always find a solution for the sum of a finite series. For example, you can add up a given series of numbers (like 1 + 2 + 3 + 4) and find the answer (10).

A finite series and infinite series only differ from each other in terms of length. You can think of an infinite series as a “… huge or enormously long series” (Lazerowitz & Ambrose, 2016).

More formally, a finite series has the form (Dragomir & Sofo, 2008):
finite series definition


  • Σ means to “sum up” (called sigma notation),
  • ai; i = 1, 2, …, n) is a sequence of numbers.

Finite Series Example: Finite Arithmetic Series

As a slightly more complicated example, the sum of the numbers 1 through 1000 is a finite sum: 500500:

This particular series is an example of an arithmetic series, which are defined by a common difference between each term (in this example, the difference is 1).

Finite Series Formulas

Some of the more common you’ll come across (Sathaye, 2020):

Arithmetic Series finite arithmetic series
Geometric Series
Telescoping Series


Dragomir, S. & Sofo, A. (2008). Advances in Inequalities for Series. Nova Science Publishers.
Lazerowitz, M. & Ambrose, A. (2016). Necessity and Language. Taylor & Francis.
Maor, E. (1991). To Infinity and Beyond. A Cultural History of the Infinite. Princeton University Press.
Riley, K. et al. (2006). Mathematical Methods for Physics and Engineering. A Comprehensive Guide.
Sathaye, A. Finite Series Formulas. Retrieved August 9, 2020 from:

Stephanie Glen. "Finite Series" From Calculus for the rest of us!

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