Dave's Math Tables:
Series Convergence Tests |

(Math | Calculus
| Expansions | Series | Convergence Tests) |

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.

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 theneither both converge or both diverge.

a _{n}anda _{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 by

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:

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 ininconclusive.

Let L = lim (n -- > ) | a_{n}|^{1/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 ininconclusive.

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.