Difference between revisions of "Alternating sum"

Revision as of 16:09, 22 August 2006

An alternating sum is a series of real numbers in which the terms alternate sign.

For example, the alternating harmonic series is $1 - \frac12 + \frac13 - \frac 14 + \ldots = \sum_{i = 1}^\infty \frac{(-1)^{i+1}}{i}$ .

Alternating sums also arise in other cases. For instance, the divisibility rule for 11 is to take the alternating sum of the digits of the integer in question and check if the result is divisble by 11.

Given an infinite alternating sum, $\sum_{i = 0}^\infty (-1)^i a_i$ , with $a_i \geq 0$ , if corresponding sequence $a_0, a_1, a_2, \ldots$ approaches a limit of zero monotonically then the series converges.

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-The alternating sum of a numbers digits, mentioned in the divisibility rule you were recently viewing, is, if the number is <math>n_0n_1n_2n_3n_4n_5...</math>, <math>n_0-n_1+n_2-n_3+n_4-n_5+...</math>.
+An '''alternating sum''' is a [[series]] of [[real number]]s in which the terms alternate sign.
-In short, it is the sum of the digits with alternating signs.
+For example, the alternating [[harmonic series]] is <math>1 - \frac12 + \frac13 - \frac 14 + \ldots = \sum_{i = 1}^\infty \frac{(-1)^{i+1}}{i}</math>.
+Alternating sums also arise in other cases.  For instance, the [[divisibility rule]] for 11 is to take the alternating sum of the [[digit]]s of the [[integer]] in question and check if the result is divisble by 11.
+Given an [[infinite]] alternating sum, <math>\sum_{i = 0}^\infty (-1)^i a_i</math>, with <math>a_i \geq 0</math>, if corresponding sequence <math>a_0, a_1, a_2, \ldots</math> approaches a [[limit]] of [[zero (constant) | zero]] [[monotonic]]ally then the series converges.
+{{stub}}

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Difference between revisions of "Alternating sum"

Revision as of 16:09, 22 August 2006