A Level Maths Notes: FP1 – Summation of Series – Telescoping Series
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.