Question Number 114099 by mnjuly1970 last updated on 17/Sep/20
$$\:\:\:….\:\:{mathematical}\:\:{analysis}….\:\: \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\:\frac{\mathrm{3}^{{n}} −\mathrm{1}}{\mathrm{4}^{{n}} }\right)\zeta\left({n}+\mathrm{1}\right)\:=\pi\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970}# \\ $$$$ \\ $$
Answered by maths mind last updated on 17/Sep/20
$$=\underset{{n}\geqslant\mathrm{1}} {\sum}.\underset{{m}\geqslant\mathrm{1}} {\sum}\left(\frac{\mathrm{3}^{{n}} −\mathrm{1}}{\mathrm{4}^{{n}} }\right).\frac{\mathrm{1}}{{m}^{{n}+\mathrm{1}} } \\ $$$$=\underset{{m}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{m}}\left\{\underset{{n}\geqslant\mathrm{1}} {\sum}\left(\frac{\mathrm{3}}{\mathrm{4}.{m}}\right)^{{n}} −\underset{{n}\geqslant\mathrm{1}} {\sum}\left(\frac{\mathrm{1}}{\mathrm{4}{m}}\right)^{{n}} \right\} \\ $$$$=\underset{{m}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{m}}\left\{\frac{\mathrm{3}}{\mathrm{4}{m}}.\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{3}}{\mathrm{4}{m}}}−\frac{\mathrm{1}}{\mathrm{4}{m}}.\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}{m}}}\right\} \\ $$$$=\underset{{m}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{m}}\left\{\frac{\mathrm{3}}{\mathrm{4}{m}−\mathrm{3}}\right\}−\underset{{m}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{m}\left(\mathrm{4}{m}−\mathrm{1}\right)} \\ $$$$=\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{3}}{\left(\mathrm{1}+{m}\right)\left(\mathrm{4}{m}+\mathrm{1}\right)}−\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}}{\left({m}+\mathrm{1}\right)\left(\mathrm{4}{m}+\mathrm{3}\right)} \\ $$$$=\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}}{\left({m}+\mathrm{1}\right)\left({m}+\frac{\mathrm{1}}{\mathrm{4}}\right)}−\underset{{m}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{1}−\frac{\mathrm{3}}{\mathrm{4}}}{\left({m}+\mathrm{1}\right)\left({m}+\frac{\mathrm{3}}{\mathrm{4}}\right)} \\ $$$$=\Psi\left(\mathrm{1}\right)−\Psi\left(\frac{\mathrm{1}}{\mathrm{4}}\right)−\left\{\Psi\left(\mathrm{1}\right)−\Psi\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right\} \\ $$$$=\Psi\left(\frac{\mathrm{3}}{\mathrm{4}}\right)−\Psi\left(\frac{\mathrm{1}}{\mathrm{4}}\right)=\Psi\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}\right)−\Psi\left(\frac{\mathrm{1}}{\mathrm{4}}\right)=\pi{cot}\left(\frac{\pi}{\mathrm{4}}\right)=\pi \\ $$
Commented by mnjuly1970 last updated on 18/Sep/20
$${thank}\:{you}\:{sir}.{grateful}… \\ $$