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# User:Jaume Oliver Lafont/The harmonic series diverges

## The harmonic series diverges

### Proof

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

Splitting into even and odd denominator terms,

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

Instead, multiplying and dividing by two,

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

${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{(2k+1)(2k+2)}}=0}$,
which is equivalent to ${\displaystyle \log(2)=\log(1)}$.