I have a question concerning the following series : $$ (H_{2}-1)+(H_{3}-1)\left(1-\frac{H_{2}}{2}\right)+(H_{4}-1)\left(1-\frac{H_{2}}{2}\right)\left(1-\frac{H_{3}}{3}\right)+(H_{5}-1)\left(1-\frac{H_{2}}{2}\right)\left(1-\frac{H_{3}}{3}\right)\left(1-\frac{H_{4}}{4}\right)+\ldots$$ Which can be rewritten as : $$ (H_{2}-1)+\sum_{n=3}^{N}\left(H_{n}-1\right)\left[\prod_{j=2}^{n-1}\left(1-\frac{H_{j}}{j}\right)\right] $$ With : $$H_{n}=\sum_{k=1}^{n}\frac{1}{k}$$ When $N$ tends to $+\infty$, this sum converges to 1. Do you have an idea to prove this ? Thank you in advance !
2026-03-30 05:28:36.1774848516
Convergence of a series implying harmonic numbers
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