The sum of first n terms of the series $$1$$⋅$$1$$!$$+2$$⋅$$2$$!$$+3$$⋅$$3$$!$$+4$$⋅$$4$$!$$+$$… is

Question

The sum of first n terms of the series $$1$$⋅$$1$$!$$+2$$⋅$$2$$!$$+3$$⋅$$3$$!$$+4$$⋅$$4$$!$$+$$… is

Infinity Learn · Accepted Answer

Let $$Sn=1$$⋅$$1$$!$$+2$$⋅$$2$$!$$+3$$⋅$$3$$!$$+4$$⋅$$4$$!  $$+$$…$$+n$$×n!⇒ $$Sn=(2$$−$$1)1$$!$$+(3$$−$$1)2$$!$$+(4$$−$$1)3$$!  $$+(5$$−$$1)4$$!$$+$$…$$+$$[(n+1)−$$1$$]n! $$=(2$$⋅$$1$$!−$$1$$!$$)+(3$$⋅$$2$$!−$$2$$!$$)+(4$$⋅$$3$$!−$$3$$!$$)$$  $$+(5$$⋅$$4$$!−$$4$$!$$)+$$…$$+$$[(n+1)n$$!−n!] $$=(2$$!−$$1$$!$$)+(3$$!−$$2$$!$$)+(4$$!−$$3$$!$$)+(5$$!−$$4$$!$$)+$$…$$+$$[$$(n+1)!−n!] $$=(n+1)$$!−$$1$$!$$=(n+1)$$!−$$1$$

The sum of first n terms of the series $1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4! + \dots is$

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The sum of first n terms of the series 1⋅1!+2⋅2!+3⋅3!+4⋅4!+… is

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The sum of first n terms of the series $1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4! + \dots is$