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  • A-Level Further Maths

Series

Sigma notation is used to write series quickly. This describes each term of a series as an equation, where substituting in values for r gives that term in the series.


There are standard sums of series formulae that are used all the time:


The Sum of Natural Numbers

To find the sum of a series of constant terms, use the formula:

Sums of series, sigma notation. Free online a-level further maths core pure 1 CP1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

To find the sum of the first n natural numbers (positive integers):

Sums of series, sigma notation. Free online a-level further maths core pure 1 CP1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

If the series does not start at r = 1, but at r = k, subtract one series from another, where the first goes from r = 1 to r = k, and the second from r = 1 to r = (k-1):

Sums of series, sigma notation. Free online a-level further maths core pure 1 CP1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

Expressions in sigma notation can be rearranged to simplify complicated series:

Sums of series, sigma notation. Free online a-level further maths core pure 1 CP1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

The Sum of Squares

The first sum formula above, for constant terms, is linear. The second, for the sum of natural numbers, is quadratic. Therefore, the formula for the sum of square terms is cubic.


To find the sum of the sum of the squares of the first n natural numbers:

Sums of series, sigma notation. Free online a-level further maths core pure 1 CP1 notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.