Consider the series

MATHEMATICS · Subjective · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

Consider the series

\ln x + p \ln x + \frac{1}{3} \ln x + \dots

, where

x \in \mathbb{R}

x > 1

and

p \in \mathbb{R}

p \neq 0

. Consider the case where the series is geometric. (a.i) Show that

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

. (a.ii) Given that

p > 0

and

S_\infty = 3 + \sqrt{3}

, find the value of

x

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

d

. (b.i) Show that

p = \frac{2}{3}

. (b.ii) Write down

d

in the form

k \ln x

, where

k \in \mathbb{Q}

. (b.iii) The sum of the first

n

terms of the series is

-3 \ln x

. Find the value of

n