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Find $$\lim_{n\to\infty} \left( \left(\sum_{k=n+1}^{2n}2\sqrt[2k]{2k}-\sqrt[k]{k}\right)-n\right).$$

I have tried rewriting the sum in a clever way, applying the Mean Value Theorem or Stolz-Cesaro Lemma somehow but haven't found anything fruitful.

Can someone please share a hint/trick to evaluate this?

Thank you.

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up vote 1 down vote accepted

Hint: use the power series expansion $$ \sqrt[y]y = \exp\bigg(\frac{\log y}y\bigg) = 1 + \frac{\log y}y + O\bigg( \frac{(\log y)^2}{y^2} \bigg) $$ (or upper and lower bounds for $\exp(\cdot)$ that express the same idea).

Reality check: the answer is a tidy number about 4% less than $\frac12$.

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