show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what interval?
2026-04-04 07:25:59.1775287559
Determine the value of r where the series converges
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Hint: To show the first thing you are asked to do, find the natural logarithm of each of the given expressions.
For the second problem, find equivalently the $r$ for which $$\sum_1^\infty \frac{1}{n^{-\ln r}}$$ converges. Put in this way, the problem should feel familiar.