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Fie \( (a_n)_{n\geq1} \) un sir crescator de numere reale pozitive. Demonstrati ca
\( \lim_{n\to\infty}\sqrt[n]{a_1^n+a_2^n+...+a_n^n}=\lim_{n\to\infty}a_n \).

\( a^n < a_1^n+a_2^n+...+a_n^n < n a_n^n \)

\( a_n <\sqrt[n]{a_1^n+a_2^n+...+a_n^n} < \sqrt[n]{n} \cdot a_n \)

Din T. Clestelui rezulta ca
\( \lim_{n\to\infty} \sqrt[n]{a_1^n+a_2^n+...+a_n^n} =\lim_{n \to \infty} a_n \)