Alternating Series

A series with terms that alternate signs.
Alternating Series Test

An alternating series converges if the absolute values of its terms are decreasing and approach zero.
Comparison Test

A series with positive terms converges if there is another series with all terms greater or equal
which is known to converge. Similarly, a series with positive terms diverges if there is another
series with all terms lesser or equal which diverges.
Convergent

The property that the partial sums of a series have a welldefined limit.
Absolutely Convergent

The property that the sum of the absolute values of the terms in a series form a convergent series.
An absolutely convergent series is automatically convergent.
Divergent

A property of a series with partial sums that do not have a welldefined limit.
Geometric Series

A series characterized by a constant ratio between consecutive terms.
Integral Test

If
f (x)
is a positive decreasing function, the series
f
_{n} = f (n)
converges if and only if the integral
f (x)dx


tends to a finite limit as
n→∞
.
Partial Sum

The sum of finitely many terms from the beginning of a series.
Power Series

A series of the form
a
_{n}
x
^{n}
where
a
_{n}
is a sequence of real numbers and
x
is a variable.
Radius of Convergence

A power series
a
_{n}
x
^{n}
converges absolutely either for all
 x < r
, or for
all real numbers
x
. We then say that the
radius of convergence of the power series is
r
or
∞
, respectively.
Ratio Test

A method for determining convergence by computing the ratios between
consecutive terms of a series. Specifically, if there is a real
number
0≤C < 1
such that
(a
_{n+1}/a
_{n})≤C
for all
n > 0
, then the series
a
_{n}
converges. This is
nothing more than the comparison test applied to a geometric
series.
Series

A sum of the elements in a sequence.
Upper Bound

A number which is greater than or equal to all of the partial sums of a sequence.