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$\lim\limits_{n \to \infty} n( \sqrt[n] {a_1}+\sqrt[n] {a_2}+...\sqrt[n] {a_p}-p) $ I am not to sure how to start . I was thinking that the p at the end repeats itself n times (-1-1-...-1) so we could assign -1 to each element of the sequence It will look something like this :$\lim\limits_{n \to \infty}\:\:\frac{( a_1^{1/n}-1+a_2^{1/n}-1+...+a_p^{1/n}-1)}{1/n} $ and after this i thought to apply the Stolz–Cesàro theorem . I would like to know if this is the way to go ...

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By L-Hospital's rule $$L=\lim_{x \rightarrow 0}\frac{a^x-1}{x}=\ln a$$ So firther from your step, you get the limit as $$\ln a_1 + \ln a_2+...+\ln a_p =\ln (a_1 a_2 a_3...a_p)$$

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