200 Followers Problem - A Logarithmic Integral!

Calculus Level 5

\[\large{L_m = \lim_{n \to \infty} \dfrac{1}{n^m} \int_1^e \prod_{k=1}^{m} \ln \left(1+ a_k x^n \right)\ \mathrm{d} x}\]

Define a sequence \((a_i)_{i=1}^\infty\) such that \(\large{a_i = e^i}\). If \(\large{L_7}\) can be represented as \(A+Be\) for integers \(A,B\), then find the value of \(A+B\).


Bonus - Generalize this problem for \(L_m, \, m \in \mathbb Z^+\). What if we change the sequence \(a_i\) as \(a_i = F_i\), where \(F_i\) denotes the \(i^{th}\) Fibonacci Number?

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