An Interesting Series

$\quad 1.1!+2.2!+3.3!+.....+n.n!\\ =(2-1)1!+(3-1)2!+(4-1)3!+...+[(n+1)-1]n!\\ =2!-1!+3!-2!+4!-3!+......+(n+1)!-n!\\ =(n+1)!-1!$

$Here,\quad the\quad value\quad of\quad n\quad is\quad 2014.\\ So,\quad the\quad answer\quad is\quad 2015.$

Here is another way. 1(1!)+2(2!)+..+2014(2014!)={2(1!)+3(2!)+...+2015(2014!)} -{1!+2!+...+2014!}={2!+3!+...+2015!}-{1!+2!+...+2014!}=2015!-1

Just hit n trial!.consider first two terms which is 1+4=5=3!-1.Now consider first three 1+4+18=23=4!-1. The last no of the series is 2014,therefore the ans is 2015

by mathematical induction we can prove that 1.1! + 2.2!+..............+ n.n! = (n+1)!-1 hence here n = 2014 so n +1 = 2015