Finding the convergence of a parametric series

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The discussion centers on the convergence of a parametric series where terms approach zero as n approaches infinity. The user struggles to determine the conditions under which the series converges, finding that neither the root criterion nor the ratio test is effective. They attempted to analyze the term ##\sqrt[n]{n}## by rewriting it as ##e^{\frac{ln n}{n}}## but encountered difficulties in proceeding further. The conclusion suggests that the parameter α must be greater than or equal to 1 for convergence. Overall, the conversation highlights the challenges in applying standard convergence tests to this specific series.
Fochina
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Homework Statement
find for what ## \alpha ## the series converges
Relevant Equations
$$\sum_{n}\left ( \sqrt[n]{n}-\sqrt[n]{2} \right )^\alpha $$
It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with ##e^{\frac{ln n}{n}}## but then I don't understand how to proceed.
 
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The result seems to be ##\alpha \geq 1##. So you could work with a convergent upper and a divergent lower bound which gives you some flexibility to change the function.
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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