Deduce related series from given series

A question is this type if and only if it asks to write down or deduce a series for a related function (e.g., ln(1-x) from ln(1+x)) without full derivation.

6 questions · Standard +0.4

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n

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Edexcel F2 2023 January Q1

8 marks Standard +0.3

Given that $y = \ln ( 5 + 3 x )$
1. determine, in simplest form, $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$
2. Hence determine the Maclaurin series expansion of $\ln ( 5 + 3 x )$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, giving each coefficient in simplest form.
3. Hence write down the Maclaurin series expansion of $\ln ( 5 - 3 x )$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, giving each coefficient in simplest form.
4. Use the answers to parts (b) and (c) to determine the first 2 non-zero terms, in ascending powers of $x$, of the Maclaurin series expansion of
$$\ln \left( \frac { 5 + 3 x } { 5 - 3 x } \right)$$

OCR FP2 Specimen Q3

7 marks Standard +0.3

Find the first three terms of the Maclaurin series for $\ln ( 2 + x )$.
Write down the first three terms of the series for $\ln ( 2 - x )$, and hence show that, if $x$ is small, then $$\ln \left( \frac { 2 + x } { 2 - x } \right) \approx x$$

AQA Further AS Paper 1 2023 June Q6

6 marks Moderate -0.3

Find and simplify the first five terms in the Maclaurin series for $\mathrm { e } ^ { 2 x }$ 6
Hence, or otherwise, write down the first five terms in the Maclaurin series for $\mathrm { e } ^ { - 2 x }$ 6
Hence, or otherwise, show that the Maclaurin series for $\cosh ( 2 x )$ is $$a + b x ^ { 2 } + c x ^ { 4 } + \ldots$$ where $a$, $b$ and $c$ are rational numbers to be determined.

AQA FP3 2006 January Q4

14 marks Standard +0.8

Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 } + \ldots$$ to write down the first four terms in the expansion, in ascending powers of $x$, of $\ln ( 1 - x )$.
The function f is defined by $$\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }$$ Use Maclaurin's theorem to show that when $\mathrm { f } ( x )$ is expanded in ascending powers of $x$ :
1. the first three terms are $$1 + x + \frac { 1 } { 2 } x ^ { 2 }$$
2. the coefficient of $x ^ { 3 }$ is zero.
Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { \sin x } - 1 + \ln ( 1 - x ) } { x ^ { 2 } \sin x }$$ (4 marks)

AQA Further Paper 2 Specimen Q15

10 marks Challenging +1.3

Show that $(1-\frac{1}{4}e^{2i\theta})(1-\frac{1}{4}e^{-2i\theta}) = \frac{1}{16}(17-8\cos 2\theta)$ [3 marks]
Given that the series $e^{2i\theta} + \frac{1}{4}e^{4i\theta} + \frac{1}{16}e^{6i\theta} + \frac{1}{64}e^{8i\theta} + \ldots$ has a sum to infinity, express this sum to infinity in terms of $e^{2i\theta}$ [2 marks]
Hence show that $\sum_{n=1}^{\infty} \frac{1}{4^{n-1}} \cos 2n\theta = \frac{16\cos 2\theta - 4}{17 - 8\cos 2\theta}$ [4 marks]
Deduce a similar expression for $\sum_{n=1}^{\infty} \frac{1}{4^{n-1}} \sin 2n\theta$ [1 mark]

WJEC Further Unit 4 2019 June Q7

6 marks Moderate -0.3

Write down the Maclaurin series expansion for $\ln(1 - x)$ as far as the term in $x^3$. [2]
Show that $-2\ln\left(\frac{1-x}{(1+x)^2}\right)$ can be expressed in the form $ax + bx^2 + cx^3 + \ldots$, where $a$, $b$, $c$ are integers whose values are to be determined. [4]