In English the expression
means sum all the terms in the series from
to
Often we have a formula for
and often the series simplifies in some way. For example a series may telescope. or collapse, with many terms cancelling
Example: Find an expression in terms of n for
(1)
All the terms cancel apart from the first and last one. Hence
In practice we may not be given the summation in the form (1). Often we have to separate the summand into partial fractions. (1) could have been given as
Example
a) Express
in partial fractions
b)Hence prove that
a)
(1)
Sub
into (1)
Sub
into (1)
b)
because this can be express as a sum of linear partial fractions some of the terms may cancel.
Very careful inspection of the terms show that all the terms cancel apart from the first and last two. hence
After some simplification this expression becomes
as required.