Let \( s_{n} \) denote the \( n \) th partial sum of the harmonic series series \( \sum_{k=1}^{\infty} \frac{1}{k} \). When \( n=2^{k-1} \), the terms of \( s_{n} \) may be bracketed as shown: \( s_{n}=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\ldots+\frac{1}{8}\right)+\cdots+\left(\frac{1}{2^{k-2}+1}+\frac{1}{2^{k-2}+2}+\ldots+\frac{1}{2^{k-1}}\right) \) Hence use an argument to show that the harmonic series diverges.
Added by Ahsan N.
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