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Calculus BC: Series

Terms

Introduction and Summary

Convergence of 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

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 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.

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