Math2.org Math Tables: Convergence Tests |
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The n^{th} partial sum of the series a_{n} is given by S_{n} = a_{1} + a_{2} + a_{3} + ... + a_{n}. If the sequence of these partial sums {S_{n}} converges to L, then the sum of the series converges to L. If {S_{n}} diverges, then the sum of the series diverges.
If a_{n} = A, and b_{n} = B, then the following also converge as indicated:
ca_{n} = cA
(a_{n} + b_{n}) = A + B
(a_{n} - b_{n}) = A - B
Absolute Convergence
If the series |a_{n}| converges, then the series a_{n} also converges.Alternating Series Test
If for all n, a_{n} is positive, non-increasing (i.e. 0 < a_{n+1} <= a_{n}), and approaching zero, then the alternating series
(-1)^{n} a_{n} and (-1)^{n-1} a_{n}
both converge.
If the alternating series converges, then the remainder R_{N} = S - S_{N} (where S is the exact sum of the infinite series and S_{N} is the sum of the first N terms of the series) is bounded by |R_{N}| <= a_{N+1}
If N is a positive integer, then the seriesboth converge or both diverge.
a_{n} and
a_{n}
n=N+1
If 0 <= a_{n} <= b_{n} for all n greater than some positive integer N, then the following rules apply:
If b_{n} converges, then a_{n} converges.
If a_{n} diverges, then b_{n} diverges.
The geometric series is given by
a r^{n} = a + a r + a r^{2} + a r^{3} + ...
If |r| < 1 then the following geometric series converges to a / (1 - r).If |r| >= 1 then the above geometric series diverges.
If for all n >= 1, f(n) = a_{n}, and f is positive, continuous, and decreasing thenLimit Comparison Testeither both converge or both diverge.
a_{n} and a_{n}
If the above series converges, then the remainder R_{N} = S - S_{N} (where S is the exact sum of the infinite series and S_{N} is the sum of the first N terms of the series) is bounded by 0< = R_{N} <= (N..) f(x) dx.
If lim (n-->) (a_{n} / b_{n}) = L,
where a_{n}, b_{n} > 0 and L is finite and positive,
then the series a_{n} and b_{n} either both converge or both diverge.
If the sequence {a_{n}} does not converge to zero, then the series a_{n} diverges.
The p-series is given byRatio Test
1/n^{p} = 1/1^{p} + 1/2^{p} + 1/3^{p} + ...
where p > 0 by definition.
If p > 1, then the series converges.
If 0 < p <= 1 then the series diverges.
If for all n, n 0, then the following rules apply:Root Test
Let L = lim (n -- > ) | a_{n+1} / a_{n} |.
If L < 1, then the series a_{n} converges.
If L > 1, then the series a_{n} diverges.
If L = 1, then the test in inconclusive.
Let L = lim (n -- > ) | a_{n} |^{1/n}.Taylor Series Convergence
If L < 1, then the series a_{n} converges.
If L > 1, then the series a_{n} diverges.
If L = 1, then the test in inconclusive.
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-->) R_{n} = 0 for all x in I.
The remainder R_{N} = S - S_{N} of the Taylor series (where S is the exact sum of the infinite series and S_{N} 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.