Summation of Series – Telescoping Series

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In English the expressionmeans sum all the terms in the series fromtoOften we have a formula forand 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) Expressin partial fractions

b)Hence prove that

a)(1)

Subinto (1)

Subinto (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 becomesas required.

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