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Determine whether the series ∑ ((1 + n2)1/2 – n) converges

by RoRi

Test whether the following series converges:

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

This series diverges.

Proof. We use the comparison test with the known divergent series \sum \frac{1}{3n}. First,

    \begin{align*} \left( n + \frac{1}{3n} \right)^2 &= n^2 + \frac{2}{3} + \frac{1}{9n^2} \\[9pt] &< n^2 + 1 \end{align*}

since \frac{1}{9n^2} < \frac{1}{3} for all n \geq 1. Therefore,

    \begin{align*} && n^2 + 1 &> \left( n + \frac{1}{3n} \right)^2 \\[9pt] \implies && \sqrt{n^2 + 1} &> n + \frac{1}{3n} \\[9pt] \implies && \sqrt{n^2 + 1} - n &> \frac{1}{3n}. \end{align*}

Thus, by the comparison test we have established the divergence of

    \[ \sum_{n=1}^{\infty} \left( \sqrt{1+n^2} - n \right). \qquad \blacksquare \]

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