|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], a [[series (mathematics)|series]] or [[integral]] is said to be '''conditionally convergent''' if it converges, but it does not [[Absolute convergence|converge absolutely]].
| | The author is known as Irwin. Years in the past we moved to North Dakota. To do aerobics is a thing that I'm completely addicted to. Supervising is my profession.<br><br>Here is my webpage: [http://Lovebyt.es/dietmealsdelivered20142 lovebyt.es] |
| | |
| ==Definition==
| |
| More precisely, a series <math>\scriptstyle\sum\limits_{n=0}^\infty a_n</math> is said to '''converge conditionally''' if
| |
| <math>\scriptstyle\lim\limits_{m\rightarrow\infty}\,\sum\limits_{n=0}^m\,a_n</math> exists and is a finite number (not ∞ or −∞), but <math>\scriptstyle\sum\limits_{n=0}^\infty \left|a_n\right| = \infty.</math>
| |
| | |
| A classic example is given by
| |
| | |
| :<math>1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum\limits_{n=1}^\infty {(-1)^{n+1} \over n}</math> | |
| | |
| which converges to <math>\ln (2)\,\!</math>, but is not absolutely convergent (see [[Harmonic series (mathematics)|Harmonic series]]).
| |
| | |
| The simplest examples of conditionally convergent series (including the one above) are the [[alternating 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 <math>\sin (x^2)</math>.
| |
| | |
| ==See also==
| |
| *[[Absolute convergence]]
| |
| *[[Unconditional convergence]]
| |
| | |
| ==References==
| |
| * Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964).
| |
| | |
| [[Category:Mathematical series]]
| |
| [[Category:Integral calculus]]
| |
| [[Category:Convergence (mathematics)]]
| |
| [[Category:Summability theory]]
| |
The author is known as Irwin. Years in the past we moved to North Dakota. To do aerobics is a thing that I'm completely addicted to. Supervising is my profession.
Here is my webpage: lovebyt.es