I don't see what happens and why this transformation is correct. $(\frac{1-z^{n+1}}{1-z}-1)\frac{1}{n} + \sum_{k=1}^{n-1} (\frac{1}{k} - \frac{1}{k+1})(\frac{1-z^{k+1}}{1-z}-1)$ = $\frac{1-z^{n+1}}{1-z}\frac{1}{n}- 1+ \sum_{k=1}^{n-1} (\frac{1}{k} - \frac{1}{k+1})\frac{1-z^{k+1}}{1-z}$
2026-04-08 04:08:06.1775621286
Reshape expression
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Note that there is a telescoping sum $$1 - \frac{1}{n} = \sum_{k=1}^{n-1} \left( \frac{1}{k} - \frac{1}{k+1} \right) $$
If you subtract this equation from your equation you will get an obvious equality.