Terms
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 well-defined 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 well-defined 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
fn = 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
anxn where
an is a sequence of real numbers and
x is a variable.
Radius of Convergence
-
A power series
anxn 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
(an+1/an)≤C for all
n > 0, then the series
an 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.
