Sum of Sequence of Factorials
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Theorem
The sequence $S = \sequence {s_n}$ defined as:
- $\ds s_n = \sum_{k \mathop = 1}^n k!$
begins:
- $1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$
This sequence is A007489 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
\(\ds s_1\) | \(=\) | \(\ds 1!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | Definition of Factorial |
\(\ds s_2\) | \(=\) | \(\ds s_1 + 2!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + 2\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 3\) |
\(\ds s_3\) | \(=\) | \(\ds s_2 + 3!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 + 6\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 9\) |
\(\ds s_4\) | \(=\) | \(\ds s_3 + 4!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 9 + 24\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 33\) |
\(\ds s_5\) | \(=\) | \(\ds s_4 + 5!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 33 + 120\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 153\) |
\(\ds s_6\) | \(=\) | \(\ds s_5 + 6!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 153 + 720\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 873\) |
\(\ds s_7\) | \(=\) | \(\ds s_6 + 7!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 873 + 5040\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 5913\) |
\(\ds s_8\) | \(=\) | \(\ds s_7 + 8!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5913 + 40 \, 320\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 46 \, 223\) |
\(\ds s_9\) | \(=\) | \(\ds s_8 + 9!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 46 \, 223 + 362 \, 880\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 409 \, 113\) |
\(\ds s_{10}\) | \(=\) | \(\ds s_9 + 10!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 409 \, 113 + 3 \, 628 \, 800\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \, 037 \, 913\) |
$\blacksquare$