Difference between revisions of "Alternating sum"
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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. | 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. | ||
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| + | ==Error estimation== | ||
| + | Suppose that an infinite alternating sum <math>\sum_{i=0}^{\infty} (-1)^ia_i</math> satisfies the the above test for convergence. Then letting <math>\sum_{i=0}^{\infty} (-1)^ia_i</math> equal <math>S</math> and the <math>k</math>-term partial sum <math>\sum_{i=0}^{k} (-1)^ia_i</math> equal <math>S_k</math>, the <b> Alternating Series Error Bound </b> states that <cmath>|S - S_k| \leq a_{k+1}.</cmath> The value of the error term <math>S - S_k</math> must also have the opposite sign as <math>(-1)^ka_k</math>, the last term of the partial series. | ||
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| + | ==Examples of infinite alternating sums== | ||
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| + | <cmath>\frac{1}{3} = \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \dots = \sum_{i=1}^{\infty} \left(-\frac{1}{2} \right)^i</cmath> | ||
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| + | <cmath>\cos 1 = 1 - \frac{1}{2} + \frac{1}{24} - \frac{1}{720} + \dots = \sum_{i=0}^{\infty} \frac{(-1)^i}{(2i)!}</cmath> | ||
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| + | <cmath>\sin 1 = 1 - \frac{1}{6} + \frac{1}{120} - \frac{1}{5040} + \dots = \sum_{i=0}^{\infty} \frac{(-1)^i}{(2i+1)!}</cmath> | ||
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| + | <cmath>\frac{1}{e} = e^{-1} = \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} + \dots = \sum_{i=2}^{\infty} \frac{(-1)^i}{i!}</cmath> | ||
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Latest revision as of 21:13, 19 February 2022
An alternating sum is a series of real numbers in which the terms alternate sign.
For example, the alternating harmonic series is 
.
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, 
, with 
, if corresponding sequence 
approaches a limit of zero monotonically then the series converges.
Error estimation
Suppose that an infinite alternating sum 
satisfies the the above test for convergence. Then letting equal 
and the 
-term partial sum 
equal 
, the Alternating Series Error Bound states that ![\[|S - S_k| \leq a_{k+1}.\]](http://latex.artofproblemsolving.com/d/8/b/d8b3d6f9896c1dc94980b98c5f072621e172c5dc.png)
The value of the error term 
must also have the opposite sign as 
, the last term of the partial series.
Examples of infinite alternating sums
![\[\frac{1}{3} = \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \dots = \sum_{i=1}^{\infty} \left(-\frac{1}{2} \right)^i\]](http://latex.artofproblemsolving.com/6/e/b/6eb4b31a4cfc84ddc4ed32c9a2c400df6282c8ef.png)
![\[\cos 1 = 1 - \frac{1}{2} + \frac{1}{24} - \frac{1}{720} + \dots = \sum_{i=0}^{\infty} \frac{(-1)^i}{(2i)!}\]](http://latex.artofproblemsolving.com/5/9/8/598a0743592ec3e1e20942a41c0b214777132fa9.png)
![\[\sin 1 = 1 - \frac{1}{6} + \frac{1}{120} - \frac{1}{5040} + \dots = \sum_{i=0}^{\infty} \frac{(-1)^i}{(2i+1)!}\]](http://latex.artofproblemsolving.com/b/e/d/bed033a139c1f366387456796011ce922e1a6416.png)
![\[\frac{1}{e} = e^{-1} = \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} + \dots = \sum_{i=2}^{\infty} \frac{(-1)^i}{i!}\]](http://latex.artofproblemsolving.com/e/f/0/ef0cf2ac6c8ec01f5e8c5222dce2687b8ffa0d7c.png)
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