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.
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.
The property that the partial sums of a series have a well-defined limit.
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.
A property of a series with partial sums that do not have a well-defined limit.
A series characterized by a constant ratio between consecutive terms.
If f (x)
is a positive decreasing function, the series fn = f (n)
converges if and only if the integral
tends to a finite limit as n→∞
The sum of finitely many terms from the beginning of a series.
A series of the form anxn
is a sequence of real numbers and
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
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
n > 0
, then the series an
converges. This is
nothing more than the comparison test applied to a geometric
A sum of the elements in a sequence.
A number which is greater than or equal to all of the partial sums of a sequence.