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


Bonus: Generalize this problem for \(L_m\), where \(m \in \mathbb Z^+\). What if \(a_i = F_i\), where \(F_i\) denotes the \(i\)th Fibonacci number?

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