inf
--- 1
\ ----
/___ ln(n)
n=2

Diverges, even though the lim

_{n->inf}1/ln(n) = 0...

**nth-Term Test for Divergence**

If a sequence {a_{n}} does not converge to 0, then the series
Sigma(a_{n}) diverges. This does **not** imply that if the sequence {a_{n}} converges to 0 that the series Sigma(a_{n}) converges; the test is merely inconclusive. In other words...

inf
--- n
\ ----
/___ 2
n=1

Diverges because lim

_{n->inf} n/2 = infinity, which does not exist.

**Geometric Series**

Geometric series are rather simple when it comes to determining their convergence or divergence. First, it would be best to define geometric series.

inf
---
\ ar^n = a + ar + ar^2 + ar^3 +... ar^n +..., r != 0
/___
n=0

The r term is referred to as the

ratio. If 0 < |r| < 1, then the geometric series converges. To determine the

series, if it converges, simply follow the formula S = a/(1-r).

**Integral Test**

This is another rather simple test that works wonderfully for all easily integrable problems. If f is a positive, continuous, decreasing function for n >= 1, and if f(n) = a_{n}, then both

|\ infinity
| f(x) dx and
\| 1
inf
---
\ a_{n}
/___
n=0

either converge or diverge.

**P-Series**

P-series are are special series defined as follows:

inf
---
\ 1/n^p = 1/1^p + 1/2^p +... 1/n^p
/___
n=1

To figure out the divergence or convergence of these series, just look at the

p. If 0 < p <= 1, then the series diverges. If p > 1, the

series converges.

**Comparison Tests**

If a_{n} <= c_{n} for all n, and

inf
---
\ c_{n} converges, then
/___
n=1
inf
---
\ a_{n} converges
/___
n=1

And if c

_{n} <= a

_{n},

inf
---
\ c_{n} diverges, then
/___
n=1
inf
---
\ a_{n} diverges
/___
n=1

In a similar light, suppose a

_{n} > 0 and d

_{n} > 0. lim

_{n->} d

_{n}/a

_{n} = L, where L is finite and positive, then the series

inf
---
\ d_{n} and
/___
n=1
inf
---
\ a_{n}
/___
n=1

both either converge or diverge.

There are several other tests for divergence and convergence, but the preceeding tests are a solid foundation for dealing with infinite series.