Here’s a nifty vid from Khan Academy proving that the harmonic series diverges: https://www.youtube.com/watch?v=4yyLfrsSXQQ.
Summary: Construct another series S with each term consisting of a power of 1/2 but less than or equal to its corresponding term in the geometric series G, i.e.:
G = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 ….
S = 1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 ….
This can be expressed as:
S = 1 + [i=1,∞]∑(1/(2^i))(2^(i-1))
Which simplifies to:
S = 1 + [i=1,∞]∑(1/2)
Which obviously diverges. Therefore, since S diverges and is less than G, the harmonic series must diverge.
Pretty clever, Mr. Oresme. But lest you feel all smug and superior, sitting there with your fancy pen in your cushy medieval office, you should know that Nicole is a girl’s name. Also, your desk is uncomfortably cubist. And you live in the 14th Century. Actually, I feel kinda bad for you. Never mind.
