to be not absolutely convergent or conditionally convergent. To this
latter class belong some convergent alternating series. For example,
the series

1 - 1/2^2 + 1/3^3 - 1/4^4 + 1/5^5 - ...

is absolutely convergent, since the series (C), p. 217, namely,

1 + 1/2^2 + 1/3^3 + 1/4^4 + 1/5^5 + ...

is convergent. The series

1 - 1/2 + 1/3 - 1/4 + 1/5 - ...

is conditionally convergent, since the harmonic series

1 + 1/2 + 1/3 + 1/4 + 1/5 + ...

is divergent.

A series with terms of different signs is convergent if the series deduced
from it by making all the signs positive is convergent.

The proof of this theorem is omitted.

Assuming that the ratio test on p. 219 holds without placing any
restriction on the signs of the terms of a series, we may summarize
our results in the following[**P3: : ?]

General directions for testing the series

u_1 + u_2 + u_3 + u_4 + ... + u_n + u_n+1 + ... .

When it is an alternating series whose terms never increase in numerical
value, and if                    limit_n=$$ u_n = 0,

then the series is convergent.

In any series in which the above conditions are not satisfied, we determine
the form of u_n and, u_n+1 and calculate the limit

limit_n=$$ (u_n+1 / u_n) = \rho.

I. When |\rho| < 1, the series is absolutely convergent.

II. When |\rho| > 1, the series is divergent.

III. When |\rho| = 1 , there is no test, and we should compare the series
with some series which we know to be convergent, as

a + ar + ar^2 + ar^3 + ...; r < 1, (geometric series)

1 + 1/2^p + 1/3^p + 1/4^p + ...; p > 1, (p series)
