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| In mathematics, the [[divergent series]]
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| :<math>\sum_{k=0}^\infty (-1)^k k!</math>
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| was first considered by [[Leonhard Euler|Euler]], who applied resummation methods to assign a finite value to the series.<ref>{{Harv|Euler|1760|p=205}}</ref> The series is a sum of [[factorial]]s that alternatingly are added or subtracted. A simple way to sum the divergent series is by using [[Borel summation]]:
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| :<math>\sum_{k=0}^\infty (-1)^k k! = \sum_{k=0}^\infty (-1)^k \int_0^\infty x^k \exp(-x) \, dx</math>
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| If we interchange summation and integration, we obtain:
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| :<math>\sum_{k=0}^\infty (-1)^k k! = \int_0^\infty \left[\sum_{k=0}^\infty (-x)^k \right]\exp(-x) \, dx</math>
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| The summation in the square brackets converges and equals 1/(1 + ''x'') if ''x'' < 1. If we replace the summation by 1/(1 + ''x'') regardless of whether it converges, we obtain a convergent integral for the summation:
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| :<math>\sum_{k=0}^\infty (-1)^{k} k! = \int_0^\infty \frac{\exp(-x)}{1+x} \, dx = e E_1 (1) \approx 0.596347362323194074341078499369\ldots</math>
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| where <math>E_1 (z)</math> is the [[exponential integral]].
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| ==Results==
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| The results for the first 10 values of ''k'' are shown below:
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| {| class="wikitable"
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| ! ''k'' !! Increment<br>calculation !! Increment !! Result
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| | 0 || 1 · 0[[Factorial|!]] = 1 · 1 || 1 || 1
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| | 1 || −1 · 1 || −1 || 0
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| | 2 || 1 · 2 · 1 || 2 || 2
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| | 3 || −1 · 3 · 2 · 1 || −6 || −4
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| | 4 ||1 · 4 · 3 · 2 · 1 || 24 || 20
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| | 5 ||−1 · 5 · 4 · 3 · 2 · 1 || −120 || −100
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| | 6 || 1 · 6 · 5 · 4 · 3 · 2 · 1 || 720 || 620
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| | 7 || −1 · 7 · 6 · 5 · 4 · 3 · 2 · 1 || −5040 || −4420
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| | 8 || 1 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 || 40320 || 35900
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| | 9 || −1 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 || −362880 || −326980
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| |}
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| ==See also==
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| * [[1 + 1 + 1 + 1 + · · ·]]
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| * [[Grandi's series]]
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| * [[1 + 2 + 3 + 4 + · · ·]]
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| * [[1 + 2 + 4 + 8 + · · ·]]
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| *[[1 − 2 + 3 − 4 + · · ·]]
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| * [[1 − 2 + 4 − 8 + · · ·]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{Citation|last=Euler|first=L.|author-link=Leonhard Euler|year=1760|title=De seriebus divergentibus|journal=Novi Commentarii academiae scientiarum Petropolitanae|issue=5|pages=205–237|publisher=|url=http://www.math.dartmouth.edu/~euler/pages/E247.html|format=PDF}}
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| {{Series (mathematics)}}
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| {{DEFAULTSORT:1 − 1 + 2 − 6 + 24 − 120 +}}
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| [[Category:Divergent series]]
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| [[Category:Series]]
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43 yr old Technical Wire Jointer Reinaldo Gomer from Matachewan, has pastimes such as golf, como ganhar dinheiro na internet and base jumping. Gains enormous inspiration from life by going to spots like Imperial Palaces of the Ming and Qing Dynasties.
Feel free to visit my web page: como conseguir dinheiro