2
$\begingroup$

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?

  • 0
    The limit then appear to be like $x^2 / log(x)^{O(log(x))}$ which is $0$.2012-10-31

1 Answers 1

1

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.

  • 0
    Agreed Lucian !2013-11-27