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2006 Mathematics for Its Sake, 3
Tags: real analysis , arithmetic sequence , arithmetic series , study convergence , sequence , limit of sequence
Let be two positive real numbers $ a,b, $ and an infinite arithmetic sequence of natural numbers $ \left( x_n \right)_{n\ge 1} . $ Study the convergence of the sequences $$ \left( \frac{1}{x_n}\sum_{i=1}^n\sqrt[x_i]{b} \right)_{n\ge 1}\text{ and } \left( \left(\sum_{i=1}^n \sqrt[x_i]{a}/\sqrt[x_i]{b} \right)^\frac{x_n}{\ln x_n} \right)_{n\ge 1} , $$ and calculate their limits. [i]Dumitru Acu[/i]