infty | |
\sum | |
n=1 |
an
If the limit of the summand is undefined or nonzero, that is
\limnan\ne0
This is also known as d'Alembert's criterion.
Suppose that there exists
r
\limn\toinfty\left|
an+1 | |
an |
\right|=r.
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.
This is also known as the nth root test or Cauchy's criterion.
Let
r=\limsupn\toinfty\sqrt[n]{|an|},
where
\limsup
infty
If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.[1]
The series can be compared to an integral to establish convergence or divergence. Let
f:[1,infty)\to\R+
f(n)=an
{an}
A commonly-used corollary of the integral test is the p-series test. Let
k>0
infty | ||
\sum | ( | |
n=k |
1 | |
np |
)
p>1
The case of
p=1,k=1
p=2,k=1
\pi2 | |
6 |
p>1,k=1
p
\zeta(p)
If the series
infty | |
\sum | |
n=1 |
bn
|an|\le|bn|
infty | |
\sum | |
n=1 |
an
If
\{an\},\{bn\}>0
\limn\toinfty
an | |
bn |
Let
\left\{an\right\}
A=
infty | |
\sum | |
n=1 |
an
A*=
infty | |
\sum | |
n=0 |
2n
a | |
2n |
A\leqA*\leq2A
Suppose the following statements are true:
\suman
\left\{bn\right\}
\left\{bn\right\}
Then
\sumanbn
Every absolutely convergent series converges.
Suppose the following statements are true:
an
\limnan=0
an+1\lean
Then
infty | |
\sum | |
n=1 |
(-1)nan
infty | |
\sum | |
n=1 |
(-1)n+1an
If
\{an\}
\{bn\}
an\geqan+1
\limnan=0
N | |
\left|\sum | |
n=1 |
bn\right|\leqM
where M is some constant, then the series
infty | |
\sum | |
n=1 |
anbn
converges.
A series
infty | |
\sum | |
i=0 |
ai
\varepsilon>0
|an+1+an+2+ … +an+p|<\varepsilon
holds for all and all .
Let
(an)n
(bn)n
(bn)n
\limn
an+1-an | |
bn+1-bn |
=l.
\limn
an | |
bn |
=l.
Suppose that (fn) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (Mn) satisfying the conditions
|fn(x)|\leqMn
n\geq1
x\inA
infty | |
\sum | |
n=1 |
Mn
infty | |
\sum | |
n=1 |
fn(x)
The ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case.
Let be a sequence of positive numbers.
Define
b | ||||
|
-1\right).
If
L=\limn\toinftybn
exists there are three possibilities:
An alternative formulation of this test is as follows. Let be a series of real numbers. Then if b > 1 and K (a natural number) exist such that
\left| | an+1 |
an |
\right|\le1-
b | |
n |
for all n > K then the series is convergent.
Let be a sequence of positive numbers.
Define
bn=lnn\left(n\left(
an | |
an+1 |
-1\right)-1\right).
If
L=\limn\toinftybn
exists, there are three possibilities:[2] [3]
Let be a sequence of positive numbers. If
an | |
an |
=1+
\alpha | |
n |
+O(1/n\beta)
\suman
Let be a sequence of positive numbers. Then:[4] [5] [6]
(1)
\suman
bn
bk(ak/ak+1)-bk+1\gec
(2)
\suman
bn
bk(ak/ak+1)-bk+1\le0
and
\sum1/bn
Let
infty | |
\sum | |
n=1 |
an
f:\R\to\R
f(1/n)=an
f''
x=0
infty | |
\sum | |
n=1 |
an
f(0)=f'(0)=0
Consider the series
Cauchy condensation test implies that is finitely convergent if
is finitely convergent. Since
infty | |
\sum | |
n=1 |
2n\left(
1 | |
2n |
\right)\alpha=
infty | |
\sum | |
n=1 |
2n-n\alpha=
infty | |
\sum | |
n=1 |
2(1-\alpha)
is a geometric series with ratio
While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let
\left\{an\right
infty | |
\} | |
n=1 |
infty | |
\prod | |
n=1 |
(1+an)
infty | |
\sum | |
n=1 |
an
0<an<1
infty | |
\prod | |
n=1 |
(1-an)
infty | |
\sum | |
n=1 |
an
This can be proved by taking the logarithm of the product and using limit comparison test.[8]