$1!+2!+3!...$ The sum of all the factorials up to a chosen positive integer $n$. This would be expressed using sigma notation. If you can show me this, that would be enough for me, for now.
If you can, show this for $(n-1)$, for the formula which I am making says, to find the $5^{th}$ harmonic number, add all the factorials from $1!$ up to $4!$, that is, the sum of all positive integer factorials from $1!$ to $(n-1)!$.
If you know how to set this up, it is gamma of two plus gamma of three plus gamma of four plus gamma of five, using the example of the $5^{th}$ harmonic number. Simply, it is $1+2+6+24$ that I am seeking to express. It is a small part of the formula.
$\endgroup$ 11 Answer
$\begingroup$It sounds like you want $$\large\sum_{k=1}^{n} k!$$ which represents the sum $1!+2!+\cdots+n!$ using sigma notation. The choice of letter $k$ is irrelevant, you are free to choose any other symbol (besides $n$, which is already being used).
In LaTeX, this is written \sum_{k=1}^{n} k!.
