6. Maclaurin and Taylor series – Edexcel Further Pure Mathematics 2 (FP2)

What students need to learn:
Third and higher order derivatives.

Derivation and use of Maclaurin series.
The derivation of the series expansion of \(e^x , \sin x, \cos x, ln (1 + x)\) and other simple functions may be required.

Derivation and use of Taylor series.
The derivation, for example, of the expansion of \(\sin x\) in ascending powers of \((x − \pi)\) up to and including the term in \((x − \pi)^3\) .

Use of Taylor series method for series solutions of differential equations.
Students may, for example, be required to find the solution in powers of \(x\) as far as the term in \(x^4\) , of the differential equation
\(\dfrac{d^2y}{dx^2} + x \dfrac{dy}{dx} + y = 0\) ,
such that \(y = 1\), \(\dfrac{dy}{d x} = 0\) at \(x = 0\).

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