let-u-n-0-sin-nx-2-x-2-6-dx-1-calculate-u-n-and-lim-u-n-n-2-find-nature-of-u-n-and-calaculate-it-3-find-nature-of-u-n-2-

Question Number 54777 by maxmathsup by imad last updated on 10/Feb/19

l e t u_{n} = \int_{0}^{\infty} \frac{s i n ({n x}^{2})}{x^{2} + 6} d x

1) c a l c u l a t e u_{n} a n d l i m u_{n} (n \to + \infty)

2) f i n d n a t u r e o f Σ u_{n} a n d c a l a c u l a t e i t .

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

Commented by Abdo msup. last updated on 11/Feb/19

1) w e h a v e u_{n} =_{x = \sqrt{6} t} \int_{0}^{\infty} \frac{s i n (6 {n t}^{2})}{6 (1 + t^{2})} \sqrt{6} d t

= \frac{1}{\sqrt{6}} \int_{0}^{\infty} \frac{s i n (6 {n t}^{2})}{1 + t^{2}} d t \Rightarrow 2 u_{n} = \frac{1}{\sqrt{6}} \int_{- \infty}^{+ \infty} \frac{s i n (6 {n t}^{2})}{1 + t^{2}} d t

\Rightarrow 2 \sqrt{6} u_{n} = I m (\int_{- \infty}^{+ \infty} \frac{e^{i 6 {n t}^{2}}}{1 + t^{2}} d t) l e t

φ (z) = \frac{e^{6 {i n z}^{2}}}{z^{2} + 1} t h e p o l e s o f φ a r e i a n d - i

r e s i d u s t h e o r e m g i v e \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e z (φ, i)

R e z (φ, i) = {l i m}_{z \to i} (z - i) φ (z) = \frac{e^{- 6 n i}}{2 i} \Rightarrow

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{e^{- 6 n i}}{2 i} = π {c o s (6 n) - i s i n (6 n)} \Rightarrow

2 \sqrt{6} u_{n} = - π s i n (6 n) \Rightarrow u_{n} = - \frac{π}{2 \sqrt{6}} s i n (6 n)

2) t h e s e q u e n c e (u_{n}) d i v e r g e s \Rightarrow Σ u_{n} d i v e r g e s

a l s o u_{n}^{2} = \frac{π^{2}}{24} {s i n}^{2} (6 n) d i v e r g e s \Rightarrow Σ u_{n}^{2} d i v e r g e s