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Question Number 80334 by ~blr237~ last updated on 02/Feb/20

l e t f \in L^{1} (R)

l e t u_{n} = \int_{a}^{b} f (t) s i n (n t) d t, v_{n} = \int_{a}^{b} \frac{f (t)}{t} s i n (n t)

1) P r o v e t h a t \lim_{n \to \infty} u_{n} = 0

2) D e d u c e i n t e r m o f a, b, f (0) t h e v a l u e o f \lim_{n \to \infty} v_{n}

Commented by abdomathmax last updated on 02/Feb/20

1) b y p a r t s u = f a n d v^{^{'}} = s i n (n t) \Rightarrow

u_{n} = {[- \frac{1}{n} c o s (n t) f (t)]}_{a}^{b} + \frac{1}{n} \int_{a}^{b} f^{^{'}} (t) c o s (n t) d t

= \frac{1}{n} {f (a) c o s (n a) - f (b) c o s (n b)}

+ \frac{1}{n} \int_{a}^{b} f^{^{'}} (t) c o s (n t) d t \Rightarrow

∣ u_{n} ∣⩽ \frac{1}{n} {f (a) + f (b)} + \frac{m (b - a)}{n} \to 0 (n \to + \infty) w i t h

m = s u p ∣ f^{^{'}} (t) ∣_{t \in [a, b]}