Question : What is the sum of the series given below?
$$\frac{1}{2} + \frac{1}{2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 5} + \cdots \frac{1}{2 \cdot 3 \cdot 5 \cdot 7 \cdot \cdots \cdot \sqrt{n}}$$
in the denominator these are the product of primes . What is the bound on this sum?
Is this $\le c\frac{1}{\sqrt{n}}$ ? where $c$ is some constant I know the given sum above is $\le \sqrt{n}/2$ but I need a tighter bound.
Question 2: Is this series converges to constant ?
The sum is $>1/2$. You can't have a bound that $\to 0$ like $c/\sqrt n$.
Answer for $2$: the series $$\sum_{n=1}^\infty\frac1{\prod_{k=1}^np_n}$$ is convergent by comparison with $\sum_{n=1}^\infty\frac1{n!}$:
$$\frac1{\prod_{k=1}^np_n}\le\frac1{n!}.$$