let-V-n-0-cos-nx-n-x-2-dx-with-n-integr-nstural-not-0-1-calculate-V-n-2-calculate-lim-n-nV-n-3-calculate-the-sum-n-0-V-n-

Question Number 54830 by maxmathsup by imad last updated on 12/Feb/19

l e t V_{n} = \int_{0}^{\infty} \frac{c o s (n x)}{n + x^{2}} d x w i t h n i n t e g r n s t u r a l n o t 0 .

1) c a l c u l a t e V_{n}

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

3) c a l c u l a t e t h e s u m \sum_{n = 0}^{\infty} V_{n}

Commented by maxmathsup by imad last updated on 13/Feb/19

1) w e h a v e V_{n} =_{x = \sqrt{n} t} \int_{0}^{\infty} \frac{c o s (n \sqrt{n} t)}{n (1 + t^{2})} \sqrt{n} d t = \frac{1}{\sqrt{n}} \int_{0}^{\infty} \frac{c o s (n \sqrt{n} t)}{1 + t^{2}} d t \Rightarrow

2 \sqrt{n} V_{n} = \int_{- \infty}^{+ \infty} \frac{c o s (n \sqrt{n} t)}{t^{2} + 1} d t = R e (\int_{- \infty}^{+ \infty} \frac{e^{i n \sqrt{n} t}}{t^{2} + 1} d t) l e t φ (z) = \frac{e^{i n \sqrt{n} z}}{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 s (φ, i) b u t φ (z) = \frac{e^{i n \sqrt{n} z}}{(z - i) (z + i)}

R e s (φ, i) = {l i m}_{z \to i} (z - i) φ (z) = \frac{e^{- n \sqrt{n}}}{2 i} \Rightarrow \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{e^{- n \sqrt{n}}}{2 i}

= π e^{- n \sqrt{n}} \Rightarrow 2 \sqrt{n} V_{n} = π e^{- n \sqrt{n}} \Rightarrow V_{n} = \frac{π}{2 \sqrt{n}} e^{- n \sqrt{n}}

2) {l i m}_{n \to + \infty} n V_{n} = {l i m}_{n \to + \infty} \frac{π}{2} \sqrt{n} e^{- n \sqrt{n}} = 0