let-f-n-x-e-nx-2e-2nx-with-x-from-0-1-calculate-0-f-n-x-dx-and-n-0-0-f-n-x-dx-2-find-S-x-n-0-f-n-x-and-0-S-x-dx-

Question Number 46610 by maxmathsup by imad last updated on 29/Oct/18

l e t f_{n} (x) = e^{- n x} - 2 e^{- 2 n x} w i t h x f r o m [0, + \infty [

1) c a l c u l a t e \int_{0}^{\infty} f_{n} (x) d x a n d \sum_{n = 0}^{\infty} (\int_{0}^{\infty} f_{n} (x) d x)

2) f i n d S (x) = \sum_{n = 0}^{\infty} f_{n} (x) a n d \int_{0}^{\infty} S (x) d x

Commented by maxmathsup by imad last updated on 05/Nov/18

1) \int_{0}^{\infty} f_{n} (x) d x = \int_{0}^{\infty} (e^{- n x} - 2 e^{- 2 n x}) d x = \int_{0}^{\infty} e^{- n x} d x - 2 \int_{0}^{\infty} e^{- 2 n x} d x

= {[- \frac{1}{n} e^{- n x}]}_{0}^{\infty} - 2 {[- \frac{1}{2 n} e^{- 2 n x}]}_{0}^{\infty} = \frac{1}{n} - 2 (\frac{1}{2 n}) = 0 \Rightarrow \sum_{n = 0}^{\infty} (\int_{0}^{\infty} f_{n} (x) d x) = 0

Commented by maxmathsup by imad last updated on 05/Nov/18

2) w e h a v e S (x) = \sum_{n = 0}^{\infty} e^{- n x} - 2 \sum_{n = 0}^{\infty} e^{- 2 n x}

= \sum_{n = 0}^{\infty} {(e^{- x})}^{n} - 2 \sum_{n = 0}^{\infty} {(e^{- 2 x})}^{n} a n d f o r x > 0 w e g e t

S (x) = \frac{1}{1 - e^{- x}} - 2 \frac{1}{1 - e^{- 2 x}}

\int S (x) d x = \int \frac{d x}{1 - e^{- x}} - 2 \int \frac{d x}{1 - e^{- 2 x}} + c b u t

\int \frac{d x}{1 - e^{- x}} = \int \frac{e^{x}}{e^{x} - 1} d x = l n ∣ e^{x} - 1 ∣

\int \frac{2 d x}{1 - e^{- 2 x}} = \int \frac{2 e^{2 x}}{e^{2 x} - 1} d x = l n ∣ e^{2 x} - 1 ∣ \Rightarrow \int S (x) d x = l n ∣ \frac{e^{x} - 1}{e^{2 x} - 1} ∣ + c

= l n ∣ \frac{1}{e^{x} + 1} ∣ + c \Rightarrow \int_{0}^{\infty} S (x) d x = {[- l n (e^{x} + 1)]}_{0}^{+ \infty} = - \infty .