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<p>Express $\mathrm{f}(x)=\frac{16-9x}{(1-3x)(4+x)}$ in partial fractions and hence obtain $\mathrm{f}(x)$ as a series in ascending powers of $x$, giving the first $3$ non-zero terms of this expansion. Find the coefficient of${{x}^{r}}$ in the expansion.<br><br></p>

Express $\mathrm{f}(x)=\frac{16-9x}{(1-3x)(4+x)}$ in partial fractions and hence obtain $\mathrm{f}(x)$ as a series in ascending powers of $x$, giving the first $3$ non-zero terms of this expansion. Find the coefficient of${{x}^{r}}$ in the expansion.



Binomial Expansion as a Maclaurin Series

General Terms and Coefficients in Power Series

Maclaurin Series and Power Series

<p>Find the coefficient of $\frac{1}{{{x}^{r}}}$ in the expansion in <strong>descending </strong>powers of $x$,</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\frac{1}{{{(1+3x)}^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{1}{{{(1+2x)}^{3}}}$</div></li></ol>

Find the coefficient of $\frac{1}{{{x}^{r}}}$ in the expansion in **descending **powers of $x$,

<p>Find the coefficient of ${{x}^{r}}$ in the expansion in ascending powers of $x$,</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\frac{1}{2-{{x}^{2}}}$</div></li><li><div class="list-item-paragraph">$\frac{2}{2-x}$</div></li></ol>

Find the coefficient of ${{x}^{r}}$ in the expansion in ascending powers of $x$,

<p>Express $\mathrm{f}\left( x \right)=\frac{6{{x}^{3}}+11{{x}^{2}}+2}{\left( {{x}^{2}}+1 \right)\left( 2x+3 \right)}$ in partial fractions.</p>
<p>Hence expand $\mathrm{f}\left( x \right)$ in ascending powers of $x$ up to and including the term in ${{x}^{3}}$, stating the range of values of $x$ for which the expansion is valid.</p>
<p>Find the coefficient of ${{x}^{2001}}$ in expansion of $\mathrm{f}\left( x \right)$.</p><p style="text-align: right;">[9]</p>

Express $\mathrm{f}\left( x \right)=\frac{6{{x}^{3}}+11{{x}^{2}}+2}{\left( {{x}^{2}}+1 \right)\left( 2x+3 \right)}$ in partial fractions.

Hence expand $\mathrm{f}\left( x \right)$ in ascending powers of $x$ up to and including the term in ${{x}^{3}}$, stating the range of values of $x$ for which the expansion is valid.

Find the coefficient of ${{x}^{2001}}$ in expansion of $\mathrm{f}\left( x \right)$.

Convergence and Validity of Power Series

Partial Fractions

JPJC Tutorial Question 10

Techniques of Integration

<p>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.</p><p style="text-align: right;">[3]</p>
<p>Find, in the simplest form, the coefficient of ${{x}^{2017}}$ in this expansion.</p><p style="text-align: right;">[2]</p>

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.

Products and Quotients of Power Series

2009 HCI Promo Q3

<p>Write down the first three terms in ascending powers of $x$ of the expansion of ${{\left( 1-x \right)}^{-2}}$. State the coefficient of ${{x}^{n}}$ and the range of values of $x$ for which the expansion is valid. Hence, or otherwise, find the value of $\sum_{n=1}^{\infty} \frac{n}{2^n}$.<span class="inline-marks">[5]</span></p>
<p style="text-align: center"></p>

Write down the first three terms in ascending powers of $x$ of the expansion of ${{\left( 1-x \right)}^{-2}}$. State the coefficient of ${{x}^{n}}$ and the range of values of $x$ for which the expansion is valid. Hence, or otherwise, find the value of $\sum_{n=1}^{\infty} \frac{n}{2^n}$.

2008 YJC P1 Q3

2009 HCI Promo P1 Q3 (modified)

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The curve $y=\mathrm{f}\left( x \right)$ passes through the point $\left( 0,81 \right)$ and has gradient given by</span></div><div class="list-item-paragraph" style="text-align: center">$\frac{\mathrm{d}y}{\mathrm{d}x}={{\left( \frac{1}{3}y-15x \right)}^{\frac{1}{3}}}$.</div><div class="list-item-paragraph">Find the first three non-zero terms in the Maclaurin series for $y$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Let $\mathrm{g}\left( x \right)=\frac{4-3x+{{x}^{2}}}{\left( 1+x \right){{\left( 1-x \right)}^{2}}}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Express $\mathrm{g}\left( x \right)$ in the form $\frac{A}{1+x}+\frac{1}{1-x}+\frac{B}{{{\left( 1-x \right)}^{2}}}$ , where $A$ and $B$ are constants to be determined.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">The expansion of $\mathrm{g}\left( x \right)$, in ascending powers of $x$, is</div><div class="list-item-paragraph" style="text-align: center">${{c}_{0}}+{{c}_{1}}x+{{c}_{2}}{{x}^{2}}+{{c}_{3}}{{x}^{3}}+...+{{c}_{r}}{{x}^{r}}+...$</div></li><li><div class="list-item-content"><span class="list-item-text">Find the values of ${{c}_{0}}$, ${{c}_{1}}$ and ${{c}_{2}}$ and show that ${{c}_{3}}=3$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Express ${{c}_{r}}$ in terms of $r$.</span></div><p style="text-align: right;">[1]</p></li></ol></li></ol>

The curve $y=\mathrm{f}\left( x \right)$ passes through the point $\left( 0,81 \right)$ and has gradient given by

Let $\mathrm{g}\left( x \right)=\frac{4-3x+{{x}^{2}}}{\left( 1+x \right){{\left( 1-x \right)}^{2}}}$.

Maclaurin Series from Differential Equations

Maclaurin Series by Successive Differentiation

Higher Derivatives by Implicit Differentiation

2019 RI P2 Q2

Techniques of Differentiation

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">find the expansion of $\frac{1}{{{\left( 1-3x \right)}^{3}}}$ in ascending powers of $x$ up to and including the ${{x}^{2}}$ term.</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the set of values of $x$ for which the series expansion is valid.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find, in terms of $n$, the coefficient of ${{x}^{n}}$ in the expansion in ascending powers of $x$ for $\frac{1}{{{\left( 1-3x \right)}^{3}}}$, leaving your answer in the simplest form.</span></div></li></ol>

find the expansion of $\frac{1}{{{\left( 1-3x \right)}^{3}}}$ in ascending powers of $x$ up to and including the ${{x}^{2}}$ term.

State the set of values of $x$ for which the series expansion is valid.

Find, in terms of $n$, the coefficient of ${{x}^{n}}$ in the expansion in ascending powers of $x$ for $\frac{1}{{{\left( 1-3x \right)}^{3}}}$, leaving your answer in the simplest form.

NYJC P1 Q2

<p>Let $\mathrm{g}\left( x \right)=\frac{5x-4{{x}^{2}}}{\left( 1+x \right){{\left( 1-2x \right)}^{2}}}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $\mathrm{g}\left( x \right)$ in the form $\frac{A}{1+x}+\frac{B}{{{\left( 1-2x \right)}^{2}}}+\frac{C}{1-2x}$, where $A$, $B$, and $C$ are constants to be determined.</span></div><p style="text-align: right;">[2]</p></li></ol>
<p>The expansion of $\mathrm{g}\left( x \right)$, in ascending powers of $x$, is</p>
<p style="text-align: center;">${{c}_{0}}+{{c}_{1}}x+{{c}_{2}}{{x}^{2}}+{{c}_{3}}{{x}^{3}}+\ldots +{{c}_{r}}{{x}^{r}}+\ldots$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of ${{c}_{0}}$, ${{c}_{1}}$, ${{c}_{2}}$, and ${{c}_{3}}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Express ${{c}_{r}}$ in terms of $r$.</span></div><p style="text-align: right;">[1]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $x$ for which the expansion is valid.</span></div><p style="text-align: right;">[2]</p></li></ol>

Let $\mathrm{g}\left( x \right)=\frac{5x-4{{x}^{2}}}{\left( 1+x \right){{\left( 1-2x \right)}^{2}}}$.

Express $\mathrm{g}\left( x \right)$ in the form $\frac{A}{1+x}+\frac{B}{{{\left( 1-2x \right)}^{2}}}+\frac{C}{1-2x}$, where $A$, $B$, and $C$ are constants to be determined.

The expansion of $\mathrm{g}\left( x \right)$, in ascending powers of $x$, is

${{c}_{0}}+{{c}_{1}}x+{{c}_{2}}{{x}^{2}}+{{c}_{3}}{{x}^{3}}+\ldots +{{c}_{r}}{{x}^{r}}+\ldots$.

Find the values of ${{c}_{0}}$, ${{c}_{1}}$, ${{c}_{2}}$, and ${{c}_{3}}$.

Find the range of values of $x$ for which the expansion is valid.

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