= Real-axis singular limit of the Riemann zeta function at one
{title2=$\lim_{p\downarrow0}p\zeta(1+p)=1$}
For $p>0$, <integral comparison> for $x^{-1-p}$ bounds its convergent positive series by $1/p\leq\sum_{n\geq1}n^{-1-p}\leq1+1/p$. Multiplication by $p$ gives $1\leq p\zeta(1+p)\leq1+p$, so the <squeeze theorem> proves the displayed real-axis limit. This establishes the leading $1/p$ divergence by real integration alone. The convergence of the series at each fixed positive $p$ follows from bounded increasing partial sums via the <monotone convergence theorem for sequences>.
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