• 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.