Finding the sum of following problem: `S_n=1.1!+2.2!+3.3!+.....+n.n!` which `(n!)=1.2.3....n`

 Finding the sum of following problem:
`S_n=1.1!+2.2!+3.3!+.....+n.n!` which `(n!)=1.2.3....n`

Solution

In order to solve kind of these problem, you need to start with the general term of the sequence: 

Exactly, the general term of our problem here is `k.k!`

We can rewrite such that: `k.k! = (k+1-1)k! =(k+1)k!-k! =(k+1)!-k!`

From that we can replace the value of k following our main problem:

If `k=1` then `1.1! = 2!-1!`

    `k=2` then `2.2! =3!-2!`

    `k=3` then `3.3! =4!-3!`

    ....................................

    .........................................    

      `k=n` then `n.n! =(n+1)!-n!`

Make the sum of side to side :
LHS : `S_n=(n+1)!-1!`

Hence: `S_n=(n+1)!-1!`

Solution by : Thin Sokkean