LEGACY Series

A-Level Further Core 1

Common Results and Basic Techniques

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A-Level Further Core 2

The Method of Difference

If the general term of a series can be describes as then

This is because when summing the for cancels with the next term’s: where so you get so every term cancels except the first and last. You can apply this to but less terms will cancel.

Maclaurin Series

The Maclaurin series of a function is an approximation for a function at x=0. (many common functions are exactly equal their Maclaurin expansion and are called analytical functions) The Maclaurin series for a function that is infinitely differentiable at is given by:

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A-Level Further Pure 1

Taylor Series

A Taylor series is simply just a Maclaurin series that is centred at some value rather than and is given by the formula:

where is the value for which the Taylor Series is centred.

This expansion only if exists and for values of for which the series converges.

Series Solutions of Differential Equations

You can use Taylor series to approximate solutions to LEGACY Differential Equations that can’t be solved with other techniques.

Suppose you have the equation and you have initial conditions Then you can calculate by substitution into . By successive differentiation of the original equation and substitution of previously found values, you can find values of Therefore the series expansion of is

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