We all know that the following harmonic series
diverges and grows very slowly!! I have seen many proofs of the result but recently found the following: S=11+12+13+14+15+16+⋯ >12+12+14+14+16+16+⋯=11+12+13+⋯=S.
In this way we see that S>S.
Can we conclude from this that S is divergent??
The proof can be made a bit more rigorous by setting
Note that an≥bn, an>bn when n is odd, and an=b2n−1+b2n.
also converges. However,
Since an≥bn and an>bn when n is odd.
Now, (3) says that
and (4) says that
These last two statements are contradictory, so the assumption that (2) converges must be false.