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_ …

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6 days ago · 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.

At first I thought this was the same as taking a Cartesian product, but he used the usual $\prod$ symbol for that further down the page, so I am inclined to believe there is some difference. Does anyone …

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 tends to $0$ …

5 days ago · Why isn't the expectation of a discrete random variable defined as $\prod_ {x\in\operatorname {Im X}} x^ {P (X=x)}$? Ask Question Asked today Modified today

Thus, if we apply Kirchhoff's theorem, we get $$\prod_ {m=1}^ {n-1} 4\sin^2 (\frac {m\pi} {n}) = n^2.$$ By taking square root and dividing both sides by $2^ {n-1}$, we get the desired formula.

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 ...

Nov 26, 2025 · Compute $$\lim_ {n \to \infty} \sqrt [n] {\prod_ {k=1}^n \left (1+ \frac {k} {n}\right)}$$ I've tried to solve it using limits of Riemann sums of the logarithm of the expression:

Jan 17, 2025 · $$ \prod_ {p \leq x} p \geq\prod_ {\sqrt {x} < p \leq x} p \geq \left (\sqrt {x}\right)^ {\pi (x) - \pi (\sqrt {x})} \ge e^ {\frac1 {2} (\frac {1} {2} x - 4 \sqrt {x})}, $$ But I have no idea what to do with this, …