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
\(\ds \) \(=\) \(\ds 4 \, 037 \, 913\)

$\blacksquare$