# Divergent series

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

## Definition

More precisely, a series $\scriptstyle\sum\limits_{n=0}^\infty a_n$ is said to converge conditionally if $\scriptstyle\lim\limits_{m\rightarrow\infty}\,\sum\limits_{n=0}^m\,a_n$ exists and is a finite number (not ∞ or −∞), but $\scriptstyle\sum\limits_{n=0}^\infty \left|a_n\right| = \infty.$

A classic example is the alternating series given by

$1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum\limits_{n=1}^\infty {(-1)^{n+1} \over n}$

which converges to $\ln (2)\,\!$, but is not absolutely convergent (see Harmonic series).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any sum at all, including ∞ or −∞; see Riemann series theorem.

A typical conditionally convergent integral is that on the non-negative real axis of $\sin (x^2)$ (see Fresnel integral).

Published on Apr 22, 2016

The Mathologer sets out to make sense of 1+2+3+ … = -1/12 and some of those other notorious, crazy-looking infinite sum identities. The starting point for this video is the famous letter that led to the discovery of self-taught mathematical genius Srinivasa Ramanujan in 1913 (Ramanujan is the subject of the movie “The man who knew infinity” that just started showing in cinemas.) Find out about how these identities come up in Ramanujan’s work, the role of “just do it” in math, the rules for adding infinite sums on Earth and other worlds, and what all this has to do with the mathematical super star the Riemann Zeta function.

If you want to watch some other videos that deal with these strange identities I recommend the following:
https://youtu.be/0Oazb7IWzbA (a Numberphile video featuring the mathematician Edward Frenkel)
https://youtu.be/XFDM1ip5HdU

Burkard

P.S.: If you know calculus and want to read up on all this some more, beyond what is readily available via the relevant Wiki pages and other internet resources, I recommend you read the last chapter of the book by Konrad Knopp, Theory and applications of infinite series, Dover books, 1990 (actually if you know German, read the extended version of this chapter in the original 1924 edition of the book).

Prof Edward Frenkel’s book Love and Math: http://amzn.to/1g6XP6j
Professor Frenkel is a mathematics professor at the University of California, Berkeley – http://edwardfrenkel.com

The Millennium Prize at the Clay Mathematics Institute: http://www.claymath.org

Number Line: http://youtu.be/JmyLeESQWGw