Problem Statement
An infinite geometric series has first term $u_1 = \ln x$ and second term $u_2 = (\ln x)^2$, where $x > 0$ and $x \neq 1$.
**(a)** [2 marks] Find the range of values of $x$ for which the sum to infinity exists.
**(b)** [4 marks] Given that the sum to infinity is $S_\infty = 3$, find the exact value of $x$.
Verified Solution & Marking Scheme
Part (a) [M1]
Common ratio $r = \frac{u_2}{u_1} = \frac{(\ln x)^2}{\ln x} = \ln x$
Part (a) [A1]
Convergence requires $|r| < 1 \implies -1 < \ln x < 1 \implies e^{-1} < x < e \implies \frac{1}{e} < x < e$ (and $x \neq 1$)
Part (b) [M1]
Formula for sum to infinity: $S_\infty = \frac{u_1}{1 - r} = \frac{\ln x}{1 - \ln x} = 3$
Part (b) [M1]
Multiply by denominator: $\ln x = 3(1 - \ln x) = 3 - 3\ln x$
Part (b) [A1]
$4\ln x = 3 \implies \ln x = \frac{3}{4}$
Part (b) [A1]
$x = e^{3/4}$