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 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.
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.
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.