Sequences & Series : sigma notation

[ introduction ][ sigma notation ] convergence ][ recurrence]






An ordered set of numbers obeying a simple rule is called a sequence.

2, 4, 6, 8, 10...

17, 22, 27, 32, 37... etc.

A series or progression is when the terms of a sequence are considered as a sum.

2 + 4 + 6 + 8 + 10 +...

17 + 22 + 27 + 32 + 37 + ... etc.


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Sigma Notation

Instead of writing long expressions like

      13 + 23 + 33 + 43 + 53 . . .+ n3

we are able to write:


which means ' the sum of all terms like m3 '

To show where a series begins and ends, numbers are placed above and below the sigma symbol. These are equal to the value of the variable, 'm' in this case, taken in order.


sigma notation top and bottom

more examples

sigma notation examples


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This concerns geometrical progressions that as the number of terms increase, the value of the sum approaches one specific number. This number is called the sum to infinity.

Look at this example. As the number of terms(n) increases, the sum of the progression( Sn )approaches the number 2.

eg of infinite geometrical series

convergent series

You can find out more about convergent series in the topic geometrical progressions.


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Recurrence is when there is some mathematical relation between consecutive terms in a sequence.

The Fibonacci series is a good example of this. The numbers of the series are made up by adding the two previous numbers.

0  + 1  + 1 + 2  + 3 +  5 +  8 +  13 +  21  . . .




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