We have n^{th} term
= \frac{2 + 4 + \cdots 2n }{n!} = \frac{n(n+1)}{n!}
So then given exprsssion
\frac{2}{1!} + \frac{2+4}{2!} + \frac{2+4+6}{3!}+ \cdots
=\sum_{n=1}^{\infty} \frac{n(n+1)}{n!} putting the valut from above
=\sum_{n=1}^{\infty} \frac{n(n-1) + 2n }{n!} writing n(n+1) s n(n-1) + 2n
=\sum_{n=1}^{\infty} (\frac{n(n-1)}{n!} + \frac{2n }{n!})
=\sum_{n=1}^{\infty} \frac{n(n-1)}{n!} + \sum_{n=1}^{\infty} \frac{2n }{n!}
=\sum_{n=2}^{\infty} \frac{n(n-1)}{n!} + \sum_{n=1}^{\infty} \frac{2n }{n!} we can convert the lowerr limit from n = 1 to n =2 because for n- 1 we have n(n-1) = 0 which can be dropped
=\sum_{n=2}^{\infty} \frac{1}{(n-2)!} + 2 \sum_{n=1}^{\infty} \frac{1 }{(n-1)!}
=\sum_{n=0}^{\infty} \frac{1}{n!} + 2\sum_{n=0}^{\infty} \frac{1 }{n!} by changing the limit
=3 \sum_{n=0}^{\infty} \frac{1}{n!}
= 3e by using expansion form for e