Visual Calculus - Series

Definition. Let {a_n} be a sequence. The n^th partial sum, s_n, is the sum

s_n = a₁ + a₂ + ... + a_n

We obtain the sequence of partial sums {s_n}.

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

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

then we write

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

is divergent.

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

Example.

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

Theorem. n^th Term Test. If the series

is convergent, then

Equivalently, if

then the series is divergent.

Examples.

Telescoping series.

Examples.