9
votes
1answer
270 views

Is there a $k$ such that $a_n=\frac{n^k!}{(n^k!!)^2}$ converges?

Lately I have been playing around with the sequence $$a_n(k) := \frac{n^k!}{(n^k!!)^2}.$$ For $k=1$, it does not look much like it converges. I don't know $k=2$ it converges, but it doesn't really ...
5
votes
4answers
483 views

Evaluating $\sum\limits_{n=2}^{\infty} \frac{1}{ GPF(n) GPF(n+1)}\,$, where $\operatorname{ GPF}(n)$ is the greatest prime factor

$\operatorname{ GPF}(n)=$Greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. $\operatorname{ LPF}(n)=$Least prime factor of $n$, eg. $\operatorname{ ...
5
votes
2answers
255 views

Convergence of $\sum\limits_{k=1}^{\infty} \frac{1}{p_{k^2}}$, where $p_k$ is the $k$th prime

Two brief questions. This seems true but I don't find it using Google. (1) Isn't $$\sum_{k=1}^{\infty} \frac{1}{p(k^2)}$$, in which $p(k^2)$ is the $k^2$th prime, known to converge? I expected to ...
8
votes
3answers
192 views

For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?

The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges. What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
3
votes
3answers
220 views

Convergence of prime series

Where can I read about convergence of series constituted of prime number such as the following: $$\sum_p \frac{1}{p (\log{p})^\alpha}\;?$$ How does convergence depend on $\alpha$?