Suppose the harmonic series converges to 
:
 |
Then rearranging the terms in the sum gives
 |
which is a contradiction.
Bernoulli's Paradox
See also
Harmonic SeriesExplore with Wolfram|Alpha
References
Boas, R. P. "Some Remarkable Sequences of Integers." Ch. 3 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 39-40, 1979.Referenced on Wolfram|Alpha
Bernoulli's ParadoxCite this as:
Weisstein, Eric W. "Bernoulli's Paradox." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernoullisParadox.html