4 problems

Let $q\geq 1$ be an integer, and let $H_k=\sum_{j=1}^k\frac{1}{j}$ denote the $k$th harmonic number. For rational numbers $\alpha_{q,2},\alpha_{q,3},\ldots,\alpha_{q,q+2}$, conside…

Sun's three conjectures. The following identities hold:

Furdui's conjecture. For all integers $k\in\mathbb{N}$, there exists $a_k\in\mathbb{Q}$ such that

Let $X$ be an infinitely dimensional Banach space, and consider the sum range of a series in $X$, namely the set of all possible limits obtained by rearranging its terms. Hadwiger'…