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Double inequality

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Papapetros Vaggelis
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Double inequality

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Post by Papapetros Vaggelis » Thu Nov 10, 2016 4:36 pm

If \(\displaystyle{n\in\mathbb{N}\,,n\geq 2}\), then prove that

\(\displaystyle{n\,\left(\sqrt[n]{n+1}-1\right)<\sum_{k=1}^{n}\dfrac{1}{k}<n\,\left(1-\dfrac{1}{\sqrt[n]{n+1}}+\dfrac{1}{n+1}\right)}\).
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