let-I-n-0-1-2-1-2t-n-e-t-dt-with-n-integr-not-0-1-prove-that-t-0-1-2-1-e-1-2t-n-1-2t-n-e-t-1-2t-n-then-find-lim-n-I-n-2-prove-that-I-n-1-1-2-n-1-I-n

Question Number 30764 by abdo imad last updated on 25/Feb/18

l e t I_{n} = \int_{0}^{\frac{1}{2}} {(1 - 2 t)}^{n} e^{- t} d t w i t h n i n t e g r n o t 0

1) p r o v e t h a t \forall t \in [0, \frac{1}{2}]

\frac{1}{\sqrt{e}} {(1 - 2 t)}^{n} ⩽ {(1 - 2 t)}^{n} e^{- t} ⩽ {(1 - 2 t)}^{n} t h e n f i n d {l i m}_{n \to \infty} I_{n}

2) p r o v e t h a t I_{n + 1} = 1 - 2 (n + 1) I_{n}

3) c a l c u l a t e I_{1}, I_{2}, a n d I_{3} .

Commented by abdo imad last updated on 01/Mar/18

1) f o r 0 ⩽ t ⩽ \frac{1}{2} w e h a v e {(1 - 2 t)}^{n} ⩾ 0 1 ⩽ e^{t} ⩽ \sqrt{e} \Rightarrow \frac{1}{\sqrt{e}} ⩽ e^{- t} ⩽ 1

\Rightarrow \frac{1}{\sqrt{e}} {(1 - 2 t)}^{n} ⩽ {(1 - 2 t)}^{n} e^{- t} ⩽ {(1 - 2 t)}^{n} \Rightarrow

\frac{1}{\sqrt{e}} \int_{0}^{\frac{1}{2}} {(1 - 2 t)}^{n} d t ⩽ \int_{0}^{\frac{1}{2}} {(1 - 2 t)}^{n} e^{- t} d t ⩽ \int_{0}^{\frac{1}{2}} {(1 - 2 t)}^{n} d t b u t

\frac{1}{\sqrt{e}} \int_{0}^{\frac{1}{2}} {(1 - 2 t)}^{n} d t = \frac{1}{\sqrt{e}} {[\frac{- 1}{2 (n + 1)} {(1 - 2 t)}^{n + 1}]}_{0}^{\frac{1}{2}} = \frac{1}{2 (n + 1) \sqrt{e}}

\Rightarrow \frac{1}{2 (n + 1) \sqrt{e}} ⩽ I_{n} ⩽ \frac{1}{2 (n + 1)} a n d i t s c l e a r t h a t

{l i m}_{n \to \infty} I_{n} = 0

2) l e t i n t e g r a t e b y p a r t s u^{^{'}} = {(1 - 2 t)}^{n} a n d v = e^{- t}

I_{n} = {[\frac{- 1}{2 (n + 1)} {(1 - 2 t)}^{n + 1} e^{- t}]}_{0}^{\frac{1}{2}} + \int_{0}^{\frac{1}{2}} \frac{- 1}{2 (n + 1)} {(1 - 2 t)}^{n + 1} e^{- t} d t

= \frac{1}{2 (n + 1)} - \frac{1}{2 (n + 1)} I_{n + 1} \Rightarrow 2 (n + 1) I_{n} = 1 - I_{n + 1} \Rightarrow

I_{n + 1} = 1 - 2 (n + 1) I_{n}

3) I_{1} = 1 - 2 I_{0} b u t I_{0} = \int_{0}^{\frac{1}{2}} e^{- t} d t = {[- e^{- t}]}_{0}^{\frac{1}{2}} = 1 - e^{\frac{- 1}{2}} = 1 - \frac{1}{\sqrt{e}}

I_{2} = 1 - 4 I_{1} = 1 - 4 (1 - \frac{1}{\sqrt{e}}) = - 3 + \frac{4}{\sqrt{e}} .

I_{3} = 1 - 6 I_{2} = 1 - 6 (- 3 + \frac{4}{\sqrt{e}}) = 19 - \frac{24}{\sqrt{e}} .