An example.

Definition. Let {an} be a sequence. The nth partial sum, sn, is the sum

sn = a1 + a2 + ... + an

We obtain the sequence of partial sums {sn}.

An interactive example illustrating the difference between the sequence of terms and the sequence of partial sums.

Definition. Given a sequence {an} and the sequence of its partial sums sn, then we say that the series

is convergent if the sequence sn is convergent and has finite limit. If

then we write

If the sequence sn is not convergent then we say that the series

is divergent.

Properties of convergent series.

Convergence of geometric series.
Discussion [Using Flash] [Using Java]


Divergence of harmonic series.
Discussion [Using Flash] [Using Java]

Theorem. nth Term Test. If the series

is convergent, then

Equivalently, if

then the series is divergent.



Telescoping series.


Drill Problems.