TITLE: Is the Medieval proof of the divergence of the Harmonic series valid?
QUESTION [5 upvotes]: As far as I understand it, the Medieval argument for the divergence of the Harmonic series:
$S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\ldots$
relies on replacing each term with the largest power of $2$ less than or equal to it, term by term, giving:
$S'=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\ldots$
and concluding, that since the $i^{\text{th}}$ element of $S'\leq$ the $i^{\text{th}}$ element of $S$ for all $i$, $S'\leq S$. But $S'=1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\ldots$, which diverges, and therefore so does $S$.
My question is, to what extent is it legitimate to argue like this, replacing each term in an infinite series with a smaller term and drawing a valid conclusion from that?
For instance if I were to argue that I had the series:
$T=2+3+4+5+\ldots$
And I then subtracted, term by term, $1$ from each element of $T$, giving:
$T'=1+2+3+4+\ldots$
Then the $i^{\text{th}}$ element of $T'\leq$ the $i^{\text{th}}$ element of $T$, as before, suggesting that $T'\leq T$, according to this argument. Except in this case you can see that $T'=T+1>T$. So arguing like this seems to lead to an absurdity.
So my question is, to what extent is element-by-element subtraction like in the Medieval Harmonic series argument legitimate? What is the difference between the Harmonic series argument and my (absurd) $T$ argument?

REPLY [6 votes]: You are almost there. If $T$ was a real number, you’d be able to conclude that $T+1<T,$ which wouldn’t make sense.
So what does that tell you about $T?$

Both proofs use the theorem:

Given two sequences, $a_n,b_n$ with $0\leq b_n\leq a_n$ for all $n,$ then if $\sum_{n=1}^{\infty} b_n$ does not converges to a real number, then neither does $\sum_{n=1}^\infty a_n.$