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A series such as

is called a *p*-series. In general, a *p*-series follows the
following form:

*p*-series are useful because of the following theorem:

Thep-series

is convergent if

p> 1 and divergent otherwise.

Unfortunately, there is no simple theorem to give us the
sum of a *p*-series. For instance, the sum of the example series is

.

If *p*=1, we call the resulting series the
harmonic series:

By the above theorem, the harmonic series does not converge.

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Copyright © 1996 Department of Mathematics, Oregon State University

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