Edexcel IAL Mathematics FP2.6.2 Maclaurin series

Practise deriving Maclaurin expansions from derivatives at zero, then combining related series in ascending powers with simplified coefficients.

Exam points

calculate derivatives at x=0 to build Maclaurin terms up to a stated power
use sign changes in odd powers to write related expansions
combine two Maclaurin series to find the first non-zero terms of a quotient expression

[Maximum number: 3]

Given that $y=e^x\sin x$

Hence determine the first 5 non-zero terms in the Maclaurin series expansion for y, giving each coefficient in simplest form.

At x=0,

y=0,\quad y'=1,\quad y''=2,\quad y'''=2,\quad y^{(4)}=0,\quad y^{(5)}=-4,\quad y^{(6)}=-8

Therefore

y=0+x+\frac{x^2}{2}\cdot2+\frac{x^3}{3!}\cdot2-\frac{x^5}{5!}\cdot4-\frac{x^6}{6!}\cdot8+\cdots

= =x+x^2+\frac{x^3}{3}-\frac{x^5}{30}-\frac{x^6}{90}+\cdots.

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