limit (1!1+2!2+3!3+..+n!n-(n+1)!)?n go to inf

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may use the following approach to solve the limit, such that:

`lim_(n->oo) (1!*1 + 2!*2 + ..... + n!*n - (n + 1)!) = lim_(n->oo) (1!(2-1) + 2!(3-1) + ... + n!(n+1 - 1) - (n+1)!)`

`lim_(n->oo) (1!(2-1) + 2!(3-1) + ... + n!(n+1 - 1) - (n+1)!) = lim_(n->oo) (2! - 1! + 3! - 2! + ... n! - (n - 1)! + (n+1)! -n! - (n+1)!) `

Reducing like terms yields:

`lim_(n->oo) (2! - 1! + 3! - 2! + ... n! - (n - 1)! + (n+1)! - n! - (n+1)!) = lim_(n->oo) (-1!) = -1`

Hence, evaluating the given limit yields `lim_(n->oo) (1!*1 + 2!*2 + ..... + n!*n - (n + 1)!) = -1.`

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