Maclaurin series for ln(a+bx)

Finding Maclaurin series for logarithmic functions of the form ln(a+bx) or ln(1/(1-bx)) using differentiation. These are straightforward applications where successive derivatives follow a clear pattern.

6 questions · Standard +0.3

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

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Edexcel FP2 2017 June Q4

10 marks Standard +0.3

4. $$y = \ln \left( \frac { 1 } { 1 - 2 x } \right) , \quad | x | < \frac { 1 } { 2 }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$
Hence, or otherwise, find the series expansion of $\ln \left( \frac { 1 } { 1 - 2 x } \right)$ about $x = 0$, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Give each coefficient in its simplest form.
Use your expansion to find an approximate value for $\ln \left( \frac { 3 } { 2 } \right)$, giving your answer
to 3 decimal places.

Edexcel FP2 Q3

7 marks Standard +0.3

3. (a) Given that $y = \ln ( 1 + 5 x ) , | x | < 0.2$, find $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
(b) Hence obtain the M aclaurin series for $\ln ( 1 + 5 x ) , | x | < 0.2$, up to and including the term in $x ^ { 3 }$.

OCR FP2 2007 January Q1

5 marks Standard +0.3

1 It is given that $\mathrm { f } ( x ) = \ln ( 3 + x )$.

Find the exact values of $f ( 0 )$ and $f ^ { \prime } ( 0 )$, and show that $f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 9 }$.
Hence write down the first three terms of the Maclaurin series for $\mathrm { f } ( x )$, given that $- 3 < x \leqslant 3$.

AQA FP3 2010 January Q2

8 marks Standard +0.3

Given that $y = \ln ( 4 + 3 x )$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Hence, by using Maclaurin's theorem, find the first three terms in the expansion, in ascending powers of $x$, of $\ln ( 4 + 3 x )$.
Write down the first three terms in the expansion, in ascending powers of $x$, of $\ln ( 4 - 3 x )$.
Show that, for small values of $x$, $$\ln \left( \frac { 4 + 3 x } { 4 - 3 x } \right) \approx \frac { 3 } { 2 } x$$

OCR Further Pure Core 1 2018 December Q3

7 marks Standard +0.3

3 You are given that $\mathrm { f } ( x ) = \ln ( 2 + x )$.

Determine the exact value of $\mathrm { f } ^ { \prime } ( 0 )$.
Show that $\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }$.
Hence write down the first three terms of the Maclaurin series for $\mathrm { f } ( x )$.

OCR Further Pure Core 1 2021 June Q2

7 marks Standard +0.3

You are given that $f(x) = \ln(2 + x)$.

Determine the exact value of $f'(0)$. [2]
Show that $f''(0) = -\frac{1}{4}$. [2]
Hence write down the first three terms of the Maclaurin series for $f(x)$. [3]