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Series Convergence Tests 
(Math | Calculus | Series Expansions | Convergence Tests)
 
Definition of Convergence and Divergence in Series
The nth partial sum of the series sum (1..inf) 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 sum an = A, and sum bn = B, then the following also converge as indicated:


sum can = cA
sum (an + bn) = A + B
sum (an - bn) = A - B

Alphabetical Listing of Convergence Tests

Absolute Convergence

If the series sum (1..inf) |an| converges, then the series sum (1..inf) 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
sum (1..inf) (-1)n an   and   sum (1..inf) (-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
sum (1..inf) an and   inf  
sum (n=N+1..inf) an  
n=N+1
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 sum (1..inf) bn converges, then sum (1..inf) an converges.
If sum (1..inf) an diverges, then sum (1..inf) bn diverges.

Geometric Series Convergence

The geometric series is given by
sum (n=0..inf) 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  
sum (1..inf) an and  integral(1..inf) an 
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 <= integral(N..inf) f(x) dx.
Limit Comparison Test
If lim (n-->inf) (an / bn) = L,
where an, bn > 0 and L is finite and positive,
then the series sum (1..inf) an and sum (1..inf) bn either both converge or both diverge.

nth-Term Test for Divergence

If the sequence {an} does not converge to zero, then the series sum (1..inf) an diverges.

p-Series Convergence
The p-series is given by
sum (1..inf) 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 not equals 0, then the following rules apply:
Let L = lim (n -- > inf) | an+1 / an |.
If L < 1, then the series sum (1..inf) an converges.
If L > 1, then the series sum (1..inf) an diverges.
If L = 1, then the test in inconclusive.
Root Test
Let L = lim (n -- > inf) | an |1/n.
If L < 1, then the series sum (1..inf) an converges.
If L > 1, then the series sum (1..inf) 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:
sum (0..inf) (1/n!) f(n)(c) (x - c)n = f(x)
if and only if lim (n-->inf) 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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