Check an approximation by its error function · source reference

Printed question

(i) Using appropriate expansions from the List of Formulae (MF26), show that the series expansion of $\ln(1+e^{2x})$ in ascending powers of $x$, up to and including the term in $x^2$, is $\ln2+x+\frac{x^2}2$. [3]

(ii) Find the set of values of $x$ such that the value for the above expansion for $\ln(1+e^{2x})$ is within $\pm0.3$ of the value of $\ln(1+e^{2x})$. [2]

(iii) Deduce an expansion for $\frac2{1+e^{-2x}}$ up to and including the term in $x$. [2]

[2018/DHS/Prelim/I/2]

Printed working

$$\begin{aligned}\ln(1+e^{2x})&=\ln[1+1+2x+(2x)^2/2!+\cdots]\\&=\ln(2+2x+2x^2+\cdots)=\ln2+\ln[1+(x+x^2+\cdots)]\\&=\ln2+(x+x^2+\cdots)-\tfrac12(x+x^2+\cdots)^2+\cdots\\&=\ln2+x+x^2/2+\cdots.\end{aligned}$$

$-0.3<\ln(1+e^{2x})-[\ln2+x+x^2/2]<0.3$. Printed GC interval $-1.55<x<1.55$ (rounded).

Differentiate: $2e^{2x}/(1+e^{2x})=1+x+\cdots$. Divide numerator/denominator by $e^{2x}$: $2/(1+e^{-2x})=1+x+\cdots$.

Teacher note 1

For $e^{2x}$, use MF27: $e^x=1+x+x^2/2!+x^3/3!+\cdots$, where $x$ is replaced by $2x$. Do not expand as $\ln(1+e^{2x})=\ln[1+1+2x+(2x)^2/2!+\cdots]=(1+2x+\cdots)-(1+2x+\cdots)^2/2+\cdots$, as each bracketed term contains a constant and terms in $x,x^2,x^3,\ldots$.

Diagram 09-08-error

Mathematical review

$1+e^{2x}=2(1+x+x^2+O(x^3))$, so the log series gives $\ln2+x+x^2/2+O(x^3)$. More precisely, $\ln(1+e^{2x})=\ln2+x+\ln\cosh x$, hence error $E(x)=\ln\cosh x-x^2/2$ is even and nonpositive. For $x>0$, $E'(x)=\tanh x-x<0$. The unique positive root of $E(x)=-0.3$ is about $1.549476$; thus $-1.549476<x<1.549476$ (about $\pm1.55$ at 3 s.f.).

Differentiating the analytic series gives $2e^{2x}/(1+e^{2x})=1+x+O(x^3)$, equivalently $2/(1+e^{-2x})=1+x+O(x^3)$.

Provenance

TJC JC2 2025 H2 Maths (9758) MSM. Native transcription checked against original visible question, working and sidebars. Question page(s): 7. Working page(s): 7. The two Series source files are byte-identical; their shared visible content is expressed once and mapped to both page sets. Original records remain private for QA; this native article has no PDF or raster worksheet dependency.