Partial Sum

A series is the sum of the terms in a sequence (not the list of terms, like a sequence). The series with a finite number of terms is a easy to find the sum of the series just by adding the terms.

However, for the series with an infinite number of terms, the summation is more difficult and complicated. The series may or may not have a finite sum.

Definition of Partial Sum

A Partial Sum is the sum of part of the sequence.

A Sequence is a set of things (usually numbers) that are in order.


This is the Sequence of even numbers from 4 onwards: {4, 8, 12, 16, 20 …}

The Partial Sum of the first 3 terms of sequence above is: 4+8+12 = 24

Partial sum of an infinite series is the sum of a finite number of consecutive terms beginning with the 1st term.

When working with infinite series, it is more helpful to examine the behavior of the partial sums.


Sigma (Σ)

Partial Sums are often written using Σ to mean “sum up”

So Σ means:

  1. Sum whatever is after the Sigma
  2. The values are shown below and above the Sigma


  1. So we sum n
  2. n goes from a to b

Partial Sums Properties

Multiplying by a ConstantPartial Sum Property Multiple by Constant 1Partial Sum Property Multiple by Constant 2nk could be k2, or k(k-7)+2, or anything
c = constant value
Adding or SubtractingPartial Sum Property Adding Substracting 1Partial Sum Property Adding Substracting 2 

Partial Sums Shortcuts

Here are some shortcuts that make the sums a lot easier.

In each case we are trying to sum from 1 to value b.

Summing 1 equals bPartial Sums Shortcuts Summing 1 equals b 1Partial Sums Shortcuts Summing 1 equals b 2
Summing the constant c equals c times bPartial Sums Shortcuts Summing Constant 1Partial Sums Shortcuts Summing Constant 2
A shortcut when summing nPartial Sums Shortcuts Summing n1 1Partial Sums Shortcuts Summing n1 2
A shortcut when summing n2Partial Sums Shortcuts Summing n2 1Partial Sums Shortcuts Summing nsss
A shortcut when summing n3Partial Sums Shortcuts Summing n3 1Partial Sums Shortcuts Summing n3 2

Learn More

Zero Product Property

General Form of Equation of a Line

Directly and Inversely Proportional

Finding an Angle in a Right Angled Triangle

Algebra Index