文档介绍：Section
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Infinite Series:
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 £© The Ratio Test
∞
In the last section we saw that we could demonstrate the convergence of a series an,
Pn=1
where an ≥ 0 for all n, by showing that an approaches 0 as n →∞ as fast as the terms
of another series with nonnegative terms which is already known to converge. Both of the
techniques developed in Section , parison test and the parison test,
proved to be very useful; however, they both suffer from the drawback of requiring that we
first find a series of known behavior which allows for the parison with the series
under consideration. In this section we shall consider another test for convergence, the ratio
test, which determines whether or not the terms of a series are approaching 0 at a rate
sufficient for the series to converge without reference to any other series. Although this test
does not require knowledge of any other series, it has the limitation of being inconclusive
in certain circumstances. Unfortunately, there is no single test for convergence which is
useful under all conditions.
The ratio test determines if the terms of a given series are approaching 0 at a rate
sufficient for convergence by considering the ratio between essive terms of the series.
Specifically, suppose an > 0 for n = 1, 2, 3, . . . and
a
lim n+1 =