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Limit of ratio to nth power

2175ExpertCalculusSequences & Series

2019 ROMANIAN MATHEMATICAL OLYMPIAD · 2019

Let $(a_n)_{n \ge 1}$ be a sequence of real positive numbers, such that the sequence $(a_{n+1} - a_n)_{n \ge 1}$ converges to a non-zero limit. Compute the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n . $$
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