Question Number 143231 by mnjuly1970 last updated on 11/Jun/21
$$.....{mathematical}\:......{Analysis}.... \\ $$$$\:{if}\:::\:\:\:\boldsymbol{\phi}\left({n}\right):=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{\mathrm{2}{n}ā\mathrm{1}} {log}\left(\mathrm{1}+{x}\right){dx} \\ $$$$\:{then}\:\:{find}\:\:{the}\:{value}\:{of}\:\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\Theta:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(ā\mathrm{1}\right)^{{n}} \boldsymbol{\phi}\left({n}\right)\: \\ $$$$\:\:\:\:\:\:.......{m}.{n} \\ $$