Telescoping logs and a rational GP ratio · source reference

Printed question

(a) A sequence $u_2,u_3,u_4,\ldots$ is such that $u_n=\ln(\frac{n-1}{n+1})$, $n\ge2$. Find in terms of $N$ an expression for $S_N=u_2+u_3+\cdots+u_{N-1}+u_N$, leaving your answer as $\ln a-f(N)$, where $a$ is positive. [3]

(b)(i) Find the set of $x$ for which $\sum_{n=1}^{\infty}(\frac{3x}{x+1})^n$ has a finite sum. [3]

(ii) Given $x=0.25$, find its value. [2]

Printed working

(a)

$$\begin{aligned}S_N&=\ln\frac13+\ln\frac24+\ln\frac35+\cdots+\ln\frac{N-2}N+\ln\frac{N-1}{N+1}\\&=\ln\frac{1\cdot2\cdot3\cdots(N-2)(N-1)}{3\cdot4\cdot5\cdots(N-1)N(N+1)}\\&=\ln\frac2{N(N+1)}=\ln2-\ln[N(N+1)].\end{aligned}$$

(b)(i) The series $(\frac{3x}{x+1})+(\frac{3x}{x+1})^2+\cdots$ is geometric, ratio $r=\frac{3x}{x+1}$. Finite sum requires $|r|<1$.

Method 1 (graphical): intersections of $y=|\frac{3x}{x+1}|$ and $y=1$ are $x=-0.25,0.5$. The graph lies below 1 between them: $-0.25<x<0.5$.

Method 2 (algebraic):

$$\left|\frac{3x}{x+1}\right|<1\Rightarrow|3x|<|x+1|\Rightarrow(3x)^2<(x+1)^2\Rightarrow9x^2<x^2+2x+1,$$ $$8x^2-2x-1<0\Rightarrow(4x+1)(2x-1)<0\Rightarrow-0.25<x<0.5.$$

(b)(ii) Method 1: at $x=0.25$, the series is $\sum_{n=1}^{\infty}(\frac35)^n=\frac35+(\frac35)^2+\cdots=\frac{3/5}{1-3/5}=\frac32$.

Method 2:

$$\frac{3x/(x+1)}{1-3x/(x+1)}=\frac{3x}{1-2x}=\frac{3(0.25)}{1-2(0.25)}=1.5.$$

Teacher notes

For $x=0.25$, this is a GP with first term $\frac35$ and common ratio $\frac35$.

Source diagrams redrawn

Printed answer key

(a) $S_N=\ln2-\ln[N(N+1)]$; (b)(i) $-0.25<x<0.5$; (b)(ii) 1.5

Mathematical review and qualifications

(a) $S_N=\ln[1\cdot2\cdots(N-1)/(3\cdot4\cdots(N+1))]=\ln[2/(N(N+1))]=\ln2-\ln[N(N+1)]$.

(b)(i) $|3x/(x+1)|<1$ is equivalent to $(4x+1)(2x-1)<0$, giving $-1/4<x<1/2$. (ii) At $x=1/4$, the ratio is $3/5$ and the first term is $3/5$, so the sum is $(3/5)/(1-3/5)=3/2$.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: SIGMA NOTATION. [2023/MI/Prelim/II/3] Question PDF pages 1; solution PDF pages 5, 6. The original files remain private QA records outside this native content.