let-A-n-1-n-1-x-2-x-x-2-dx-n-integr-1-1-calculate-A-n-2-find-lim-n-A-n-

Question Number 39120 by math khazana by abdo last updated on 02/Jul/18

l e t A_{n} = \int_{1}^{n} \frac{[\sqrt{1 + x^{2}}] - [x]}{x^{2}} d x (n i n t e g r ⩾ 1)

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 math khazana by abdo last updated on 04/Jul/18

w e h a v e A_{n} = \sum_{k = 1}^{n} \int_{k}^{k + 1} \frac{[\sqrt{1 + x^{2}}] - k}{x^{2}} d x b u t

k ⩽ x < k + 1 \Rightarrow k^{2} ⩽ x^{2} < {(k + 1)}^{2} \Rightarrow

k^{2} + 1 ⩽ 1 + x^{2} ⩽ 1 + {(k + 1)}^{2} \Rightarrow \sqrt{1 + k^{2}} ⩽ \sqrt{1 + x^{2}} < \sqrt{1 + {(k + 1)}^{2}}

\Rightarrow \sqrt{1 + k^{2}} ⩽ \sqrt{1 + x^{2} <} \sqrt{k^{2} + 2 k + 2} ⩽ k + 2 \Rightarrow

k ⩽ \sqrt{1 + x^{2} <} k + 2 \Rightarrow [\sqrt{1 + x^{2}}] = k o r k + 1

A_{n} = \sum_{k = 1}^{n} \int_{k}^{k + 1} \frac{k + 1 - k}{x^{2}} d x = \sum_{k = 1}^{n} \int_{k}^{k + 1} \frac{d x}{x^{2}}

= \sum_{k = 1}^{n} {[- \frac{1}{x}]}_{k}^{k + 1} = \sum_{k = 1}^{n} (\frac{1}{k} - \frac{1}{k + 1})

= 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \dots . \frac{1}{n} - \frac{1}{n + 1} = 1 - \frac{1}{n + 1} \Rightarrow

A_{n} = \frac{n}{n + 1}

2) i t s c l e a r t h a t {l i m}_{n \to + \infty} A_{n} = 1 .

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Jul/18

x \sqrt{1 + x^{2}} [\sqrt{1 + x^{2}}] [x_{}] [\sqrt{1 + x^{2}}] - [x]

1 \sqrt{2} = 1.41 1 1 0

1.5 \sqrt{3.25} = 1.8 1 1 0

2 \sqrt{5} = 2.23 2 2 0

2.5 \sqrt{7.25} = 2.69 2 2 0

3 \sqrt{10} = 3.16 3 3 0

4 \sqrt{17} = 4.12 4 4 0

k \sqrt{1 + k^{2}} = k + △ k k k 0

w h e n 1 > △ k > 0

w a i t

Commented by abdo.msup.com last updated on 03/Jul/18

b u t y o u k n o w t h a t t h e t r a i n d o n t w a i t \dots

Commented by tanmay.chaudhury50@gmail.com last updated on 03/Jul/18

Commented by tanmay.chaudhury50@gmail.com last updated on 03/Jul/18

f (x) = [\sqrt{1 + x^{2}}] - [x]

f r o m g r a p h i t i s c l e a r t h a t f (x) = [\sqrt{1 + x^{2}}] - [x]

i s z e r o w h e n n > x ⩾ 1

s o t h e v a l u e o f i n t r e g a l i s z e r o \dots p l s c h e c k

Commented by math khazana by abdo last updated on 03/Jul/18

m i r a c y l o u s g r a p h f o r t h i s f u n c t i o n i t s l i k e

a r a i l w a y s \dots