Sum of Sequence of Factorials

Theorem

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

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

begins:

$1, 3, 9, 33, 153, 873, 5913, 46 \, 233, 409 \, 113, 4 \, 037 \, 913, \ldots$

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 $\displaystyle$ $=$ $\displaystyle 4 \, 037 \, 913$

$\blacksquare$