The Harmonic Series

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A special type of p-series is the harmonic series. This is a p-series with p=1, in other words the series

    \[ \sum_{n=1}^{\infty}\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \dots\]

Since p-series only converge when p>1, the harmonic series diverges.

Notice that the harmonic series is an example of a series whose terms tend to zero (\lim_{n\rightarrow\infty}\frac{1}{n} = 0) but the series doesn’t converge. Remember this if you’re ever tempted to use the divergence test to conclude that a series converges.

Also note that related series like

    \[ \sum_{n=1}^{\infty} \frac{3}{n} \]

and

    \[ \sum_{n=1}^{\infty} \frac{-1}{n} \]

also diverge. You can show this using the integral test, for example.