let-U-n-0-n-e-x-2-dx-2-0-n-e-nx-2-dx-1-calculate-lim-n-U-n-2-determne-nature-of-U-n-and-U-n-3-

Question Number 53464 by maxmathsup by imad last updated on 22/Jan/19

l e t U_{n} = \frac{{(\int_{0}^{n} e^{- x^{2}} d x)}^{2}}{\int_{0}^{n} e^{- {n x}^{2}} d x}

1) c a l c u l a t e {l i m}_{n \to + \infty} U_{n}

2) d e t e r m n e n a t u r e o f Σ U_{n} a n d Σ U_{n}^{3} .

Commented by maxmathsup by imad last updated on 24/Jan/19

w e h a v e \int_{0}^{n} e^{- {n x}^{2}} d x =_{\sqrt{n} x = t} \int_{0}^{n \sqrt{n}} e^{- t^{2}} \frac{d t}{\sqrt{n}} = \frac{1}{\sqrt{n}} \int_{0}^{n \sqrt{n}} e^{- t^{2}} d t \Rightarrow

U_{n} = \sqrt{n} \frac{{(\int_{0}^{n} e^{- x^{2}} d x)}^{2}}{\int_{0}^{n \sqrt{n}} e^{- t^{2}} d t} b u t {l i m}_{n \to + \infty} \sqrt{n} = + \infty

{l i m}_{n \to + \infty} \frac{{(\int_{0}^{n} e^{- x^{2}} d x)}^{2}}{\int_{0}^{n \sqrt{n}} e^{- t^{2}} d t} = \frac{{(\frac{\sqrt{π}}{2})}^{2}}{\frac{\sqrt{π}}{2}} = \frac{\sqrt{π}}{2} \Rightarrow {l i m}_{n \to + \infty} U_{n} = + \infty

2) w e h a v e U_{n} \sim \frac{\sqrt{π n}}{2} \Rightarrow Σ U_{n} d i v e r g e s a l s o Σ U_{n}^{3} d i v e r g e s .

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19

I_{1} = \int_{0}^{\infty} e^{- x^{2}} d x a n d I_{2} = \int_{0}^{\infty} e^{- {n x}^{2}} d x

c a l c u l a t i n g I_{2}

\int_{0}^{\infty} e^{- {n x}^{2}} d x

t = {n x}^{2} x = \frac{\sqrt{t}}{\sqrt{n}} \frac{d x}{d t} = \frac{1}{\sqrt{n}} \times \frac{1}{2} \times t^{\frac{1}{2} - 1} = \frac{1}{2 \sqrt{n}} \times t^{\frac{- 1}{2}}

\int_{0}^{\infty} e^{- t} \times \frac{1}{2 \sqrt{n}} \times t^{\frac{- 1}{2}} d t

= \frac{1}{2 \sqrt{n}} \int_{0}^{\infty} e^{- t} \times t^{\frac{1}{2} - 1} d t

= \frac{1}{2 \sqrt{n}} \times ⌈ (\frac{1}{2}) = \frac{1}{2 \sqrt{n}} \times \sqrt{π} = \frac{1}{2} \times \sqrt{\frac{π}{n}} \to v a l u e o f I_{2}

I_{1} = \frac{1}{2} \times \sqrt{π}

\lim_{n \to \infty} U_{n} = \frac{I_{1}^{2}}{I_{2}} = \frac{\frac{π}{4}}{\frac{1}{2} \times \sqrt{\frac{π}{n}}} = \frac{π \times 2 \times \sqrt{n}}{4 \times \sqrt{π}} = \frac{\sqrt{n π}}{2}

s i r p l s c h e 4 k \dots