sum to infinity

**righteous818** · December 15th 2012, 07:13 PM

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.

**richard1234** · December 15th 2012, 07:22 PM

$\sum_{r = 1}^{\infty} \frac{1}{r!} = e-1$

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

Adding, we get

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

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