The divergence test

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If a series diverges, it means that the sum of an infinite list of numbers is not finite (it may approach $\pm \infty$ or it may oscillate), and:

The standard example of a sequence whose terms go to zero, and yet does not converge, is the harmonic series. The Harmonic sequence, $(1/n)$ , converges to $0$ while the Harmonic Series, $\sum _{n=1}^\infty \frac {1}{n}\qquad \text {diverges.}$

Restating this point again (because it is very important): passing the divergence test means that a series has a chance to converge. The divergence test cannot tell us whether a series converges.

Some questions

It’s a great idea at this point to stop and compare the previous two questions.

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