Calculus BC: Series

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 f _n = f (n) converges if and only if the integral

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.