These pages list several series which are important for comparison purposes. They are useful for the comparison tests: the `regular' Comparison Test and the Limit Comparison Test. Click on the name of the series to get more information on the series.

*a*+*ar*+*ar*^2 + ... The term of the series
is
*ar*^*n*. This series converges if |*r*|<1 and diverges
otherwise. If it converges, it converges to *a*/(1-*r*).

The term of the *p*-series is 1/*n*^*p*. This
series
converges if *p*>1 and diverges otherwise. If *p*=1, this
is the harmonic series.

This kind of series cancels itself out. For instance if you write
out the sum over *n* from 1 to infinity of
(1/*n*-1/(*n*+1)), you'll get 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4
+ ..., and all of the terms except the "1" in front will cancel each other
out, so the series will converge to 1.

Copyright © 1996 Department of Mathematics, Oregon State University

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