# 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?

×