find-lim-x-0-1-cos-x-cos-2-2x-cos-3-3x-cos-n-nx-x-2-

Question Number 133830 by metamorfose last updated on 24/Feb/21

f i n d \lim_{x \to 0} \frac{1 - c o s (x) {c o s}^{2} (2 x) {c o s}^{3} (3 x) \dots {c o s}^{n} (n x)}{x^{2}} = \dots ?

Answered by EDWIN88 last updated on 24/Feb/21

\lim_{x \to 0} \frac{1 - (1 - \frac{1}{2} x^{2}) (1 - \frac{8}{2} x^{2}) (1 - \frac{27}{2} x^{2}) \dots (1 - \frac{n^{3}}{2} x^{2})}{x^{2}} =

\lim_{x \to 0} \frac{1 - (1 - \frac{1}{2} x^{2} (1^{3} + 2^{3} + 3^{3} + \dots + n^{3}))}{x^{2}} =

\frac{1}{2} (1^{3} + 2^{3} + 3^{3} + \dots + n^{3}) = \frac{1}{2} {[\frac{n (n + 1)}{2}]}^{2}

Answered by Dwaipayan Shikari last updated on 24/Feb/21

\lim_{x \to 0} \frac{1 - c o s (x) {c o s}^{2} (2 x) {c o s}^{3} (3 x) \dots {c o s}^{n} (n x)}{x^{2}} = y

\underset{n = 1}{\sum^{n}} n l o g (c o s (n x)) = l o g (1 - x^{2} y) \lim_{x \to 0} l o g (1 + x) = x

\underset{n = 1}{\sum^{n}} n l o g (1 - \frac{n^{2} x^{2}}{2}) = l o g (1 - x^{2} y) \lim_{x \to 0} c o s x = 1 - \frac{x^{2}}{2}

- \underset{n = 1}{\sum^{n}} \frac{n^{3} x^{2}}{2} = - x^{2} y \Rightarrow \frac{1}{2} \underset{n = 1}{\sum^{n}} n^{3} = y

y = \frac{n^{2} {(n + 1)}^{2}}{8}

Answered by bramlexs22 last updated on 24/Feb/21

\lim_{x \to 0} \frac{1 - \cos x \cos^{2} 2 x}{x^{2}} =

\lim_{x \to 0} \frac{1 - \cos x {(1 - 2 \sin^{2} x)}^{2}}{x^{2}} =

\lim_{x \to 0} \frac{1 - (1 - \frac{1}{2} x^{2}) (1 - {4 x}^{2})}{x^{2}} =

\lim_{x \to 0} \frac{1 - (1 - \frac{9}{2} x^{2} + \frac{1}{2} x^{4})}{x^{2}} = \frac{9}{2}