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. Show that

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

. b. Given that

p>0

and

S_\infty=3+\sqrt{3}

, find the value of

x

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

d

. Show that

p=\frac{2}{3}

. d. Write down

d

in the form

k\ln x

, where

k\in\mathbb{Q}

. e. The sum of the first

n

terms of the series is

-3\ln x

. Find the value of

n