Challenging +1.8 This question requires converting logarithms to natural logs using the change of base formula, recognizing the resulting geometric series structure with ratio -1/2, and applying the infinite GP sum formula. While it involves multiple conceptual steps (logarithm manipulation, series identification, and algebraic simplification), each step follows standard A-level techniques. The main challenge is the unfamiliar base pattern (powers of powers of 2) and connecting the final answer to ln(2√2), but this is within reach for strong FM students with careful algebraic work.
In the question you must show detailed reasoning
Given that \(\log_a x = \frac{\log_n x}{\log_n a}\), show that the sum of the infinite series, where \(n = 0,1,2...\),
$$\log_2 e - \log_4 e + \log_{16} e - \cdots + (-1)^n \log_{2^{2^n}} e + \cdots$$
is
$$\frac{1}{\ln(2\sqrt{2})}$$
[5]
[Total marks: 65]
In the question you must show detailed reasoning
Given that $\log_a x = \frac{\log_n x}{\log_n a}$, show that the sum of the infinite series, where $n = 0,1,2...$,
$$\log_2 e - \log_4 e + \log_{16} e - \cdots + (-1)^n \log_{2^{2^n}} e + \cdots$$
is
$$\frac{1}{\ln(2\sqrt{2})}$$
[5]
[Total marks: 65]
\hfill \mbox{\textit{SPS SPS FM 2019 Q12 [5]}}