Question Number 12414 by FilupS last updated on 21/Apr/17
$$\mathrm{does}\:\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}{p}_{{i}} \:\:\:\mathrm{converge},\:\:\:\:{p}_{{i}} \in\mathbb{P} \\ $$
Answered by prakash jain last updated on 22/Apr/17
$$\mathrm{There}\:\mathrm{are}\:\mathrm{infinitely}\:\mathrm{many}\:\mathrm{primes}. \\ $$$$\mathrm{So}\:\mathrm{the}\:\mathrm{series}\:\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}{p}_{{i}} \:\mathrm{diverges}. \\ $$
Commented by FilupS last updated on 22/Apr/17
$$\mathrm{what}\:\mathrm{about}\:\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{p}_{{i}} }\:?? \\ $$
Commented by prakash jain last updated on 22/Apr/17
$$\frac{\mathrm{1}}{{p}_{{i}} }\:\mathrm{also}\:\mathrm{diverges},\:\mathrm{same}\:\mathrm{series}\:\mathrm{as}\:\frac{\mathrm{1}}{{n}} \\ $$
Commented by FilupS last updated on 22/Apr/17
$$\mathrm{why}\:\mathrm{is}\:\mathrm{that}?\:\mathrm{can}\:\mathrm{you}\:\mathrm{show}\:\mathrm{me}\:\mathrm{why}? \\ $$