 Solution

For fixed $p>0$, the partial sums are increasing and bounded above by $1+1/p$. The monotone convergence theorem for sequences therefore proves convergence to a finite sum $S(p)$. Passing to the limit $N\to \infty$ in the two bounds givesThe squeeze theorem now yieldsSince $S(p)=\zeta (1+p)$, this is the real-axis singular limit of the Riemann zeta function at one. The limiting sum is taken at fixed $p$ before sending $p$ to zero; setting $p=0$ inside the convergent-series argument would instead give the divergent harmonic series.