Question Number 118304 by mnjuly1970 last updated on 16/Oct/20
$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\underset{\blacktrinagledown} {\overset{\blacktrinagledown} {\blacklozenge}}\mathscr{A}{dvanced}\:\:{calculus}\underset{\blacktrinagledown} {\overset{\blacktrinagledown} {\blacklozenge}}\:... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{3}^{{n}} {n}^{\mathrm{3}} }\:\overset{???} {=}\frac{\mathrm{7}}{\mathrm{8}}\zeta\left(\:\mathrm{3}\:\right)+\frac{\mathrm{1}}{\mathrm{6}}{ln}^{\mathrm{3}} \left(\mathrm{2}\right)−\frac{\pi^{\mathrm{2}} }{\mathrm{12}}\:{ln}\left(\mathrm{2}\right)\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\spadesuit\mathscr{M}.\mathscr{N}.\mathscr{J}{uly}\:\mathrm{1970}\spadesuit... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.\mathscr{G}{ood}\:{luck}. \\ $$$$ \\ $$