Infinite series divergence (nth term) test and integral test of n/(1+n^2).

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An example where the nth term is zero but series diverges.

We apply the divergence (nth term) test to the series sum(n/(1+n^2)), and the series passes the test (the terms go to zero as n goes to infinity). However, this does *not* show that the series converges! We proceed to test the convergence of the infinite series by using the integral test, and it turns out the infinite series actually diverges, despite having passed the nth term test.