# sum to infinity

• December 15th 2012, 08:13 PM
righteous818
sum to infinity
how do i find the sum to infinity of the sum of (2r+1)/r! where r>=1. i have no clue about sums and factorials.
• December 15th 2012, 08:22 PM
richard1234
Re: sum to infinity
$\sum_{r = 1}^{\infty} \frac{1}{r!} = e-1$

$2 \sum_{r = 1}^{\infty} \frac{r}{r!} = 2 \sum_{r = 1}^{\infty} \frac{1}{(r-1)!} = \sum_{r = 0}^{\infty} \frac{1}{r!} = e$ (simply shifted the "index" by 1)

$\sum_{r = 1}^{\infty} \frac{2r+1}{r!} = 2e + (e-1) = 3e - 1$