User:Jaume Oliver Lafont/The harmonic series diverges

The harmonic series diverges

Proof

Assume $\sum _{k=0}^{\infty }{\frac {1}{k+1}}=S$

Splitting into even and odd denominator terms,

$S=\sum _{k=0}^{\infty }{\frac {1}{2k+1}}+{\frac {1}{2k+2}}$ [eq. 1]

Instead, multiplying and dividing by two,

$S=\sum _{k=0}^{\infty }{\frac {1}{2k+2}}+{\frac {1}{2k+2}}$ [eq. 2]

Subtracting [eq. 1] from [eq. 2] leads to the contradiction

$\sum _{k=0}^{\infty }{\frac {1}{(2k+1)(2k+2)}}=0$,

which is equivalent to $\log(2)=\log(1)$.

This is essentially equivalent to one of the proofs in http://faculty.prairiestate.edu/skifowit/htdocs/harmapa.pdf.