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Infinite Product $\prod\limits_ {k=1}^\infty\left ( {1-\frac {x^2} {k^2 . . . 29 I've been looking at proofs of Euler's Sine Expansion, that is $$ \frac {\sin\left (x\right)} {x} = \prod_ {k = 1}^ {\infty} \left (1-\frac {x^ {2}} {k^ {2}\pi^ {2}}\right) $$ All the proofs seem to rely on Complex Analysis and Fourier Series Is there any more elementary proof ?
Finding Value of the Infinite Product $\\prod \\Bigl(1-\\frac{1}{n^{2 . . . Here is a hint to evaluate $$\prod_ {n=2}^\infty\left (1-\frac1 {n^2}\right) :$$ Note that this is a telescoping product, since $1-1 n^2= (n-1) (n+1) n^2$ Now play with the first few terms to see the emerging pattern Share Cite edited Jan 19, 2011 at 18:34 answered Jan 19, 2011 at 18:27 Andrés E Caicedo 82 2k10233368 Add a comment
trigonometry - Prove that $\prod_ {k=1}^ {n-1}\sin\frac {k \pi} {n . . . Incidentally, the proof given in fiktor's answer below can be modified to show that sin nx = 2n−1∏n−1 k=0 sin(x + kπ n) sin n x = 2 n − 1 ∏ k = 0 n − 1 sin (x + k π n), a very pretty multiple-angle identity which is not as widely know as it deserves to be Dividing by sin x sin x and letting x → 0 x → 0 reduces that identity to the one in the question
summand is to sum $\\left(\\sum\\right)$ as ___ is to product . . . Language evolves The specific words you allude to are often-times considered un-useful and unnecessary It is far more common in modern language to simply refer to them with generic language as "terms in a product" or "terms in a sum "