find-u-n-0-e-n-x-cos-nx-dx-and-v-n-0-e-n-x-sin-nx-dx-2-find-nature-of-u-n-v-n-and-u-n-v-n-

Question Number 45976 by maxmathsup by imad last updated on 19/Oct/18

f i n d u_{n} = \int_{0}^{\infty} e^{- n [x]} c o s (n x) d x a n d v_{n} = \int_{0}^{\infty} e^{n [x]} s i n (n x) d x

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

Commented by maxmathsup by imad last updated on 20/Oct/18

w e h a v e u_{n} + {i v}_{n} = \int_{0}^{\infty} e^{- n [x]} e^{i n x} d x = \int_{0}^{\infty} e^{- n [x] + i n x} d x

= \sum_{k = 0}^{\infty} \int_{k}^{k + 1} e^{- k n + i n x} d x

= \sum_{k = 0}^{\infty} e^{- n k} \int_{k}^{k + 1} e^{i n x} d x = \sum_{k = 0}^{\infty} e^{- n k} \frac{1}{i n} {[e^{i n x}]}_{k}^{k + 1}

= \frac{1}{i n} \sum_{k = 0}^{\infty} e^{- n k} {e^{i n (k + 1)} - e^{i n k}}

= \frac{1}{i n} e^{i n} \sum_{k = 0}^{\infty} e^{- n k + i n k} - \frac{1}{i n} \sum_{k = 0}^{\infty} e^{- n k + i n k}

= \frac{1}{i n} e^{i n} \sum_{k = 0}^{\infty} {(e^{- n + i n})}^{k} - \frac{1}{i n} \sum_{k = 0}^{\infty} {(e^{- n + i n})}^{k}

= \frac{1}{i n} e^{i n} \frac{1}{1 - e^{- n + i n}} - \frac{1}{i n} \frac{1}{1 - e^{- n + i n}} = \frac{e^{i n} - 1}{i n} \frac{1}{1 - e^{- n + i n}}

= \frac{c o s (n) + i s i n (n) - 1}{i n} \frac{1}{1 - e^{- n} (c o s n + i s i n (n))}

= \frac{- i (c o s (n) + i s i n (n) - 1)}{n} \frac{1}{1 - e^{- n} c o s (n) - {i e}^{- n} s i n (n)}

= \frac{s i n (n) - i c o s (n) + i}{n} \frac{1 - e^{- n} c o s (n) + i e^{- n} s i n (n)}{{(1 - e^{- n} c o s (n))}^{2} + e^{- 2 n} {s i n}^{2} (n)}

a f t e r w e s e p a r a t e R e (. .) a n d I m (\dots) \dots