let-u-n-0-dt-1-t-n-find-nature-of-u-n-and-u-n-n-2-and-u-n-n-3-

Question Number 48068 by maxmathsup by imad last updated on 18/Nov/18

l e t u_{n} = \int_{0}^{\infty} \frac{d t}{1 + t^{n}}

f i n d n a t u r e o f Σ u_{n} a n d Σ \frac{u_{n}}{n^{2}} a n d Σ \frac{u_{n}}{n^{3}}

Commented by Abdo msup. last updated on 19/Nov/18

c h a n g e m e n t t^{n} = x g i v e t = x^{\frac{1}{n}} \Rightarrow

u_{n} = \int_{0}^{\infty} \frac{1}{1 + x} \frac{1}{n} x^{\frac{1}{n} - 1} d x = \frac{1}{n} \int_{0}^{\infty} \frac{x^{\frac{1}{n} - 1}}{1 + x} d x

= \frac{1}{n} \frac{π}{s i n (\frac{π}{n})} \Rightarrow u_{n} = \frac{π}{n s i n (\frac{π}{n})}

2) w e h a v e u_{n} \sim \frac{π}{n . \frac{π}{n}} \Rightarrow u_{n} \to 1 (n \to + \infty) s o Σ u_{n} d i v e r g e

b e c a u s e u_{n} d o n t c o n v e r g e t o 0) a l s o

\frac{u_{n}}{n^{2}} \sim \frac{1}{n^{2}} a n d Σ \frac{1}{n^{2}} c o n v e r g e \Rightarrow Σ \frac{u_{n}}{n^{2}} c o n v e r g e

\frac{u_{n}}{n^{3}} \sim \frac{1}{n^{3}} a n d ? Σ \frac{1}{n^{3}} c o n v e r g e s \Rightarrow Σ \frac{u_{n}}{n^{3}} c o n v e r g e s .

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

t^{n} = {t a n}^{2} θ {n t}^{n - 1} d t = 2 t a n θ {s e c}^{2} θ d θ

\int_{0}^{\frac{π}{2}} \frac{2 t a n θ {s e c}^{2} θ d θ}{{s e c}^{2} θ \times n {({t a n}^{2} θ)}^{\frac{n - 1}{n}}}

= \frac{2}{n} \int_{0}^{\frac{π}{2}} {(t a n θ)}^{1 - \frac{2 n - 2}{n}} d θ

\frac{2}{n} \int_{0}^{\frac{π}{2}} \frac{{(s i n θ)}^{\frac{2 - n}{n}}}{{(c o s θ)}^{\frac{2 - n}{n}}} d θ

\frac{2}{n} \int {(s i n θ)}^{\frac{2 - n}{2}} \times {(c o s θ)}^{\frac{n - 2}{n}} d θ

2 p - 1 = \frac{2 - n}{2} p = \frac{4 - n}{4} = (1 - \frac{n}{4})

2 q - 1 = \frac{n - 2}{2} 2 q = \frac{n}{2} q = \frac{n}{4}

n o w f o r m u l a 2 \int_{0}^{\frac{π}{2}} {(s i n θ)}^{2 p - 1} {(c o s θ)}^{2 q - 1} d θ

= \frac{⌈ (p) ⌈ (q)}{⌈ (p + q)}

= \frac{1}{n} \times \frac{⌈ (1 - \frac{n}{4}) ⌈ (\frac{n}{4})}{⌈ (1)}

= \frac{1}{n} \times \frac{π}{s i n (\frac{n π}{4})} = \frac{π}{n} \times \frac{1}{s i n (\frac{n π}{4})}

s o u_{n} = \frac{π}{n} \times \frac{1}{s i n (\frac{n π}{4})}