Consider the series

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Consider the series

\ln x+p\ln x+\frac13\ln x+\dots

, where

x\in\mathbb{R}

x>1

, and

p\in\mathbb{R}

p\ne0

. Consider first the case where the series is geometric. (1) Show that

p=\pm\frac{1}{\sqrt{3}}

. (2) Given that

p>0

and

S_\infty=3+\sqrt{3}

, find the value of

x

. (3) Now consider the case where the series is arithmetic with common difference

d

. Show that

p=\frac{2}{3}

. (4) Write down

d

in the form

k\ln x

, where

k\in\mathbb{Q}

. (5) The sum of the first

n

terms of the series is

-3\ln x

. Find the value of

n