How the following series
$$1 - \sum_{j=2}^{n} \frac{(2-\eta)(\eta-1)}{ (j+\eta-3)(j+\eta-2) } = \frac{n(\eta-1)}{2-(2-\eta)}, \qquad \eta\neq 1,$$
has this form $\frac{n(\eta-1)}{2-(2-\eta)}$
2026-04-22 11:45:02.1776858302
Series estimate
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Hint: Use fraction decomposition \begin{align} 1 - \sum_{j=2}^{n} \frac{(2-\eta)(\eta-1)}{(j+\eta-3)(j+\eta-2)} &= 1 - (2-\eta)(\eta-1)\sum_{j=2}^{n} \frac{1}{(j+\eta-3)(j+\eta-2)}\\ &= 1 - (2-\eta)(\eta-1)\sum_{j=2}^{n} \left(\frac{1}{j+\eta-3}-\frac{1}{j+\eta-2}\right) \end{align} and telescopic property on summation.