let-A-n-0-n-e-x-x-dx-1-calculate-A-n-2-find-lim-n-A-n-

Question Number 38101 by maxmathsup by imad last updated on 21/Jun/18

l e t A_{n} = \int_{0}^{n} e^{x - [x]} d x

1) c a l c u l a t e A_{n}

2) f i n d {l i m}_{n \to + \infty} A_{n}

Commented by prof Abdo imad last updated on 22/Jun/18

A_{n} = \sum_{k = 0}^{n - 1} \int_{k}^{k + 1} e^{- k} e^{x} d x

= \sum_{k = 0}^{n - 1} e^{- k} (e^{k + 1} - e^{k})

= \sum_{k = 0}^{n - 1} (e - 1) = (e - 1) \sum_{k = 0}^{n - 1} (1) = n (e - 1)

A_{n} = n (e - 1)

2) w e h a v e e - 1 > 0 \Rightarrow {l i m}_{n \to + \infty} A_{n} = + \infty .