TITLE: $\sum\limits_{n=1}^{10000000000000000} \frac{1}{n}$
QUESTION [8 upvotes]: How does wolfram alpha solve $$\sum\limits_{n=1}^{10000000000000000} \frac{1}{n}\approx 37.4186$$so quickly? It solved it in like 3 seconds is there a equation or something

REPLY [2 votes]: we can use the prime numbers in geometric series as follow
$$1+1/2+1/3+1/4+1/5+1/6+1/7+...=1+1/2+1/4+1/8+1/16+...1/3+1/9+1/27+...+1/5+1/25+1/125+...$$
$$1+\frac{1/2}{1-1/2}+\frac{1/3}{1-1/3}+\frac{1/5}{1-1/5}+....$$