Why does harmonic series diverge?

Question

A series is said to be convergent if $$\left|\lim_{n \to \infty}\frac{U_{n+1}}{U_n}\right| \leq1$$ where $U_n$ denotes $n$th term of the series. If we denote the sum upto $j$th term of the Harmonic series by $H_j$, then $$H_j=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{j}$$ where $j\geq1$. There's a formal proof to show that this series diverges, but from the ratio test of convergent series, we get $$\left|\lim_{n \to \infty}\frac{U_{n+1}}{U_n}\right|=\left|\lim_{n \to \infty}\frac{\frac{1}{n+1}}{\frac{1}{n}}\right|=\lim_{n \to \infty}\frac{n}{n+1}=\lim_{n \to \infty}\frac{1}{1+\frac{1}{n}}=1$$ which clearly shows that this is a convergent series. What am I missing here? It's impossible for a series to be both convergent and divergent at the same time. Please provide me some insight to this problem.

Answer 1

Your first display is incorrect. The ratio test guarantees convergence if the inequality in your first display is strict. It makes no statement about convergence/divergence when the limit of the ratio of successive terms is $1$.

Answer 2

It's not Cauchy. Consider, given $N$, $|S_{2N}-S_N|=1/(2N)+\dots+1/(N+1)\ge N\cdot 1/(2N)=1/2$.