Question Number 225391 by Tawa11 last updated on 24/Oct/25
$$\:\:\:\:\:\:\:\underset{\mathrm{n}\:\:\:=\:\:\:\mathrm{1}} {\overset{\infty} {\sum}}\:\left(−\:\:\:\:\mathrm{1}\right)^{\mathrm{n}\:\:\:\:−\:\:\:\mathrm{1}} \:\frac{\mathrm{H}_{\mathrm{n}} \:\mathrm{H}_{\mathrm{2n}} ^{\left(\mathrm{2}\right)} }{\mathrm{n}}\:\:\:\:\:\:\:=\:\:\:\:\:? \\ $$