Riemann_Series_Rearrangement

Convergence of a conditionally convergent series.

Sequence

A sequence is a list of things (usually numbers, functions, etc. ) that are in order.
When the sequence goes on forever it is called an infinite sequence otherwise finite sequence.

Series

A series is the sum of the terms of an infinite sequence of numbers.
More precisely, an infinite sequence {a₁, a₂, a₃, …..} defines a series S that is denoted by

where a_k is the kth term of the series.

nth Partial Sum

Sum of the first “n” terms of the series is known as the nth partial sum and is denoted by

Convergent and Divergent Series

An infinite series is said to be converging if

"L" must be finite and unique real number. Any series that is not convergent is said to be divergent.

Absolutely Convergent Series

If the series

converges, then the series

is absolutely convergent.

Conditionally Convergent Series

If the series

converges, but the series

diverges, then the series

is conditionally convergent.

Riemann Rearrangement Theorem

Riemann rearrangement theorem, says that if an infinite series of real numbers is conditionally convergent,
then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges.

Statement of the Theorem

Suppose that

is conditionally convergent. Let M be a real number. Then there exists a permutation σ such that

There also exists a permutation σ such that

The sum can also be rearranged to diverge to -∞ or to fail to approach any limit, finite or infinite.