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MathCalculus

Integral Test for Convergence: Does ∑ n/(3n² + 1) Converge?

The positive decreasing extension has a divergent logarithmic improper integral, so ∑ n/(3n² + 1) diverges by the integral test.

Question

Use the integral test to determine whether the series

∑n=1∞n3n2+1\sum_{n=1}^{\infty}\frac{n}{3n^2+1}

converges or diverges.

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Evidence boundary

This conclusion uses the positive, continuous, decreasing extension f(x) = x/(3x² + 1) on [1, ∞). The integral test determines convergence or divergence; it does not assign a finite sum to this divergent series.