 # Maclaurin and Power Series

Maclaurin series refers to the expansion of a series of functions where the estimated value of the function is determined as a sum of the derivatives of that function when evaluated at zero. It is named after Colin Maclaurin (1698–1746), a Scottish mathematician who made extensive use of this special case of Taylor series in the 18th century.
Maclaurin series are very frequently used to approximate functions.

##### 2009 HCI Promo P1 Q3 Modified

Expand $\frac{{{x}^{2}}+2x}{2{{x}^{2}}+1}$ in ascending powers of $x$ up to and including the term in ${{x}^{5}}$. State the range of values of $x$ for which this expansion is valid.



Find, in the simplest form, the coefficient of ${{x}^{2017}}$ in this expansion.



##### Suggested Handwritten and Video Solutions    ##### 2020 MI P1 Q6

It is given that $y=\sqrt{{{\text{e}}^{x}}\cos x}$.

(i)

Show that $2y\frac{\text{d}y}{\text{d}x}={{y}^{2}}-{{\text{e}}^{x}}\sin x$.



(i) Show that $2y\frac{\text{d}y}{\text{d}x}={{y}^{2}}-{{\text{e}}^{x}}\sin x$.



(ii)

By further differentiation of the result in part (i), find the Maclaurin series for $y$, up to and including the term in ${{x}^{2}}$.



(ii) By further differentiation of the result in part (i), find the Maclaurin series for $y$, up to and including the term in ${{x}^{2}}$.



(iii)

Using the standard series from the List of Formulae (MF 26). Expand $\sqrt{{{\text{e}}^{x}}\cos x}$ as far as the term in ${{x}^{2}}$ and verify that the same result is obtained in part (ii).



(iii) Using the standard series from the List of Formulae (MF 26). Expand $\sqrt{{{\text{e}}^{x}}\cos x}$ as far as the term in ${{x}^{2}}$ and verify that the same result is obtained in part (ii).



##### Suggested Handwritten and Video Solutions      ##### 2013 ACJC P1 Q12
Given that $y={{({{\sin }^{-1}}x)}^{2}}$, show that

$\left( 1-{{x}^{2}} \right){{\left( \frac{\text{d}y}{\text{d}x} \right)}^{2}}=4y$

and $\left( 1-{{x}^{2}} \right)\frac{{{\text{d}}^{\text{2}}}y}{\text{d}{{x}^{2}}}-x\frac{\text{d}y}{\text{d}x}=2$.



By further differentiation of these results, find the Maclaurin series of $y$ up to including the term in ${{x}^{4}}$.



##### Suggested Handwritten and Video Solutions   ##### 2017 PJC Promo Q9 (a)

In the triangle $ABC$ as shown below, $BC=3$, angle $BAC=\frac{\pi }{3}+\theta$ radians and angle $ACB=\frac{\pi }{2}$ radians. Show that $AC=\frac{3\left( 1-\sqrt{3}\tan \theta \right)}{\sqrt{3}+\tan \theta }$.



Given that $\theta$ is a sufficiently small angle, deduce that $AC\approx \sqrt{3}+a\theta$, where $a$ is a constant to be determined.



##### Suggested Handwritten and Video Solutions    ##### How to show ${{\text{e}}^{\text{i}\theta }}=\cos \theta +\text{i}\sin \theta$ using Standard Series

Using the standard series in MF26, show ${{\text{e}}^{\text{i}\theta }}=\cos \theta +\text{i}\sin \theta$.

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