for-each-n-N-f-n-x-nx-1-x-2-n-for-each-x-0-x-1-and-a-n-0-1-f-n-x-dx-if-s-n-sin-pia-n-for-each-n-N-then-li-n-m-s-n-

Question Number 11089 by suci last updated on 11/Mar/17

f o r e a c h n \in N, f_{n} (x) = n x {(1 - x^{2})}^{n}

f o r e a c h x, 0 ⩽ x ⩽ 1 a n d a_{n} = \int_{0}^{1} f_{n} (x) d x

i f s_{n} = s i n (π a_{n}), f o r e a c h n \in N, t h e n

l \underset{n \to\sim}{i m} s_{n} = \dots . ? ? ?

Commented by FilupS last updated on 11/Mar/17

n \in N

f_{n} (x) = n x {(1 - x^{2})}^{n} 0 ⩽ x ⩽ 1

a_{n} = \int_{0}^{1} f_{n} (x) d x

s_{n} = \sin (π a_{n})

\lim_{n \to \infty} s_{n} = L

L = ?

a_{n} = \int_{0}^{1} f_{n} (x) d x

a_{n} = \int_{0}^{1} n x {(1 - x^{2})}^{n} d x

= n \int_{0}^{1} x {(1 - x^{2})}^{n} d x

u = 1 - x^{2}

d u = - 2 x d x

= - n \frac{1}{2} \int_{0}^{1} u^{n} d u

= - n \frac{1}{2} {(\frac{1}{n + 1} u^{n + 1})}_{0}^{1}

= - \frac{n}{2 (n + 1)} {(u^{n + 1})}_{0}^{1}

= - \frac{n}{2 (n + 1)} {({(1 - x^{2})}^{n + 1})}_{0}^{1}

= - \frac{n}{2 (n + 1)} (0^{n + 1} - 1^{n + 1})

= \frac{n}{2 (n + 1)}

∴ a_{n} = \frac{n}{2 (n + 1)}

s_{n} = \sin (π a_{n})

s_{n} = \sin (π \frac{n}{2 (n + 1)})

s_{n} = \sin (\frac{1}{2} π \frac{n}{(n + 1)})

\lim_{n \to \infty} s_{n} = \underset{n \to \infty}{\lim s i n} (\frac{1}{2} π \frac{n}{(n + 1)})

= \sin (\frac{1}{2} π [\lim_{n \to \infty} \frac{n}{(n + 1)}])

L^{'} H {\hat{o p i t a l}}^{'} s Law

= \sin (\frac{1}{2} π \lim_{n \to \infty} \frac{1}{1})

= \sin (\frac{1}{2} π)

∴ L = 1

\lim_{n \to \infty} s_{n} = 1

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