 | Resources |  | | · Cool Tools · Formulas & Tables · References · Test Preparation · Study Tips · Wonders of Math | | Search |  | | | | Series Convergence Tests | | (Math | Calculus | Series Expansions | Convergence Tests) | Definition of Convergence and Divergence in Series The nth partial sum of the series  an is given by Sn = a1 + a2 + a3 + ... + an. If the sequence of these partial sums {Sn} converges to L, then the sum of the series converges to L. If {Sn} diverges, then the sum of the series diverges. Operations on Convergent Series If  an = A, and bn = B, then the following also converge as indicated: can = cA (an + bn) = A + B (an - bn) = A - B Alphabetical Listing of Convergence Tests Absolute Convergence If the series |an| converges, then the series an also converges. Alternating Series Test If for all n, an is positive, non-increasing (i.e. 0 < an+1 <= an), and approaching zero, then the alternating series (-1)n an and (-1)n-1 an both converge. If the alternating series converges, then the remainder RN = S - SN (where S is the exact sum of the infinite series and SN is the sum of the first N terms of the series) is bounded by |RN| <= aN+1 Deleting the first N Terms If N is a positive integer, then the series both converge or both diverge. Direct Comparison Test If 0 <= an <= bn for all n greater than some positive integer N, then the following rules apply: If bn converges, then an converges. If an diverges, then bn diverges. Geometric Series Convergence The geometric series is given by  a rn = a + a r + a r2 + a r3 + ... If |r| < 1 then the following geometric series converges to a / (1 - r). If |r| >= 1 then the above geometric series diverges. Integral Test If for all n >= 1, f(n) = an, and f is positive, continuous, and decreasing then either both converge or both diverge. If the above series converges, then the remainder RN = S - SN (where S is the exact sum of the infinite series and SN is the sum of the first N terms of the series) is bounded by 0< = RN <=  (N.. ) f(x) dx. Limit Comparison Test If lim (n-->) (an / bn) = L, where an, bn > 0 and L is finite and positive, then the series an and bn either both converge or both diverge. nth-Term Test for Divergence If the sequence {an} does not converge to zero, then the series an diverges. p-Series Convergence The p-series is given by 1/np = 1/1p + 1/2p + 1/3p + ... where p > 0 by definition. If p > 1, then the series converges. If 0 < p <= 1 then the series diverges. Ratio Test If for all n, n  0, then the following rules apply: Let L = lim (n -- > ) | an+1 / an |. If L < 1, then the series an converges. If L > 1, then the series an diverges. If L = 1, then the test in inconclusive. Root Test Let L = lim (n -- > ) | an |1/n. If L < 1, then the series an converges. If L > 1, then the series an diverges. If L = 1, then the test in inconclusive. Taylor Series Convergence If f has derivatives of all orders in an interval I centered at c, then the Taylor series converges as indicated: (1/n!) f(n)(c) (x - c)n = f(x) if and only if lim (n-->) RN = 0 for all x in I. The remainder RN = S - SN of the Taylor series (where S is the exact sum of the infinite series and SN is the sum of the first N terms of the series) is equal to (1/(n+1)!) f(n+1)(z) (x - c)n+1, where z is some constant between x and c. |
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