Question Number 164425 by Zaynal last updated on 17/Jan/22
$$\mathrm{What}\:\mathrm{it}'\mathrm{s}\:?? \\ $$$$\mathrm{5}×\mathrm{55}×\mathrm{555}×\mathrm{5555}×\mathrm{55555}………….\infty= \\ $$
Answered by alephzero last updated on 17/Jan/22
$$\mathrm{Let}\:{a}_{{n}} \:=\:\underset{{n}-\mathrm{times}} {\underbrace{\mathrm{5555}…\mathrm{5555}}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}{a}_{{n}} \:=\:\mathrm{5}×\mathrm{55}×\mathrm{555}×…×\infty\:=\:? \\ $$$$\mathrm{Let}\:\underset{\mathrm{Without}\:\infty} {\underbrace{\mathrm{5}×\mathrm{55}×\mathrm{555}×…}}\:=\:{t}\:\Rightarrow\:{t}\:>\:\mathrm{0} \\ $$$$\Rightarrow\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}{a}_{{n}} \:=\:{t}×\infty\:=\:\infty \\ $$$$\forall{t}\:\left\{{t}\:\in\:\mathbb{R}\:\mid\:{t}×\infty\:=\:\infty\right\} \\ $$$$\Rightarrow\:\mathrm{5}×\mathrm{55}×\mathrm{555}×…×\infty\:=\:\underline{\infty} \\ $$