Let $n$ be a positive integer and let $p(n)$ be the $n$th prime. Let $f(n) = \dfrac{1}{30} \prod_{3
How does $f(n)$ behave asymptotically? Does $\lim_{n\to oo} (n+7)^2 f(n)$ exist and what value is it? Can the limit be given in closed form?
Let $n$ be a positive integer and let $p(n)$ be the $n$th prime. Let $f(n) = \dfrac{1}{30} \prod_{3
How does $f(n)$ behave asymptotically? Does $\lim_{n\to oo} (n+7)^2 f(n)$ exist and what value is it? Can the limit be given in closed form?
An infinite product is said to converge if the limit exists, and it is not zero. This is because the log of a product is a sum of logs, and $\ln0=-\infty$. Furthermore, there's a theorem about the convergence or divergence of $\prod_i(1-a_i)$ being the same that of $\sum_ia_i$. In this case, since $\sum_i\frac{2\,i}{p(i)\ln p(i)}$ diverges then so does the product. Since each term is clearly $<1$, then it diverges towards $0$. QED.