Sum of Sequence of Factorials

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Theorem

The sequence $S = \sequence {s_n}$ defined as:

$\displaystyle s_n = \sum_{k \mathop = 1}^n k!$

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

\(\displaystyle s_1\) \(=\) \(\displaystyle 1!\)
\(\displaystyle \) \(=\) \(\displaystyle 1\) Definition of Factorial


\(\displaystyle s_2\) \(=\) \(\displaystyle s_1 + 2!\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + 2\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 3\)


\(\displaystyle s_3\) \(=\) \(\displaystyle s_2 + 3!\)
\(\displaystyle \) \(=\) \(\displaystyle 3 + 6\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 9\)


\(\displaystyle s_4\) \(=\) \(\displaystyle s_3 + 4!\)
\(\displaystyle \) \(=\) \(\displaystyle 9 + 24\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 33\)


\(\displaystyle s_5\) \(=\) \(\displaystyle s_4 + 5!\)
\(\displaystyle \) \(=\) \(\displaystyle 33 + 120\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 153\)


\(\displaystyle s_6\) \(=\) \(\displaystyle s_5 + 6!\)
\(\displaystyle \) \(=\) \(\displaystyle 153 + 720\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 873\)


\(\displaystyle s_7\) \(=\) \(\displaystyle s_6 + 7!\)
\(\displaystyle \) \(=\) \(\displaystyle 873 + 5040\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 5913\)


\(\displaystyle s_8\) \(=\) \(\displaystyle s_7 + 8!\)
\(\displaystyle \) \(=\) \(\displaystyle 5913 + 40 \, 320\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 46 \, 223\)


\(\displaystyle s_9\) \(=\) \(\displaystyle s_8 + 9!\)
\(\displaystyle \) \(=\) \(\displaystyle 46 \, 223 + 362 \, 880\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 409 \, 113\)


\(\displaystyle s_{10}\) \(=\) \(\displaystyle s_9 + 10!\)
\(\displaystyle \) \(=\) \(\displaystyle 409 \, 113 + 3 \, 628 \, 800\) Definition of Factorial
\(\displaystyle \) \(=\) \(\displaystyle 4 \, 037 \, 913\)

$\blacksquare$