Dec 28, 2013 · 21 The symbol $\Pi$ is the pi-product. It is like the summation symbol $\sum$ but rather than addition its operation is multiplication. For example, $$ \prod_ …

DevOps engineers are those who are good at debugging, troubleshooting, analyzing prod issues and providing solutions. Who have good hands on technologies like unix shell scripting, perl, …

4 days ago · Consider the product $$ P = \prod_ {k=1}^ {12} \left (1 + \sqrt {\omega^k - 1}\right), $$ where a consistent branch of the square root is chosen. Is there a closed form for $P$?

Dec 24, 2025 · One way, I guess to see this, is that this procedure fixes $\prod_ {i=1}^nx_i$, and when taking the logarithm is equivalent to the averaging process. Thus, we get the result.

Sep 10, 2024 · By L'Hospital: The derivative of the denominator is (by pulling one cosine at a time from the product) $$\sum_ {i=1}^n\frac {i\sin (ix)} {\cos (ix)}\prod_ {i=1}^n\cos (ix).$$ This still …

May 8, 2014 · 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 …

Oct 23, 2024 · This is not a duplicate of Closed form of $\int\limits_0^ {2\pi} \prod\limits_ {j=1}^n \cos (jx)dx$ and combinatorial link as the following does not apply: As suggested by Winther in …

Nov 1, 2024 · There are simple reasons for the others - it is that $1$ and $4$ are squares of integers.

Dec 12, 2025 · We have $$\begin {align*} L &= \lim_ {N\to\infty} \prod_ {n=1}^ {N} \frac {\frac {\pi} {2}\arctan (n)} {\arctan (2n-1)\arctan (2n)} \\ &= \lim_ {N\to\infty} \prod_ {n ...

@DanPetersen: The friend said "the terms in the product" - that is, the numbers being multiplied together - have values less than $1$, and therefore the value of the product can never be $1$. …