Question Number 91960 by frc2crc last updated on 04/May/20
$$\mathrm{If} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{an}\:{e}\mathrm{ven}\:\mathrm{function}\:\mathrm{is} \\ $$$$\underset{{n}=−\infty} {\overset{\infty} {\sum}}{f}\left({n}\right)=\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{f}\left({n}\right)\:{true}? \\ $$
Commented by abdomathmax last updated on 04/May/20
$$\sum_{{n}=−\infty} ^{+\infty} \:{f}\left({n}\right)\:=\sum_{{n}=−\infty} ^{\mathrm{0}} \:{f}\left({n}\right)\:+\sum_{\mathrm{0}} ^{\infty} \:{f}\left({n}\right) \\ $$$${changement}\:{of}\:\:{indice}\:{m}=−{n}\:{give} \\ $$$$\sum_{{n}=−\infty} ^{\mathrm{0}} \:=\sum_{{m}=\mathrm{0}} ^{+\infty} \:{f}\left(−{m}\right)\:=\sum_{{m}=\mathrm{0}} ^{+\infty\:} {f}\left({m}\right)\:\Rightarrow \\ $$$$\sum_{{n}=−\infty} ^{+\infty} \:{f}\left({n}\right)\:=\mathrm{2}\sum_{{n}=\mathrm{0}} ^{\infty} \:{f}\left({n}\right)\:{the}\:{relstion}\:{is}\:{true} \\ $$$${but}\:\:{we}\:{must}\:{took}\:{a}\:{look}\:{to}\:{convergence}... \\ $$
Commented by frc2crc last updated on 04/May/20
$${thanks} \\ $$
Commented by abdomathmax last updated on 09/May/20
$${you}\:{are}\:{welcome} \\ $$