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 | ||
$$\mathrm{Let}\:{a}_{{n}} \:=\:\underset{{n}-\mathrm{times}} {\underbrace{\mathrm{5555}...\mathrm{5555}}} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}{a}_{{n}} \:=\:\mathrm{5}×\mathrm{55}×\mathrm{555}×...×\infty\:=\:? \\ $$$$\mathrm{Let}\:\underset{\mathrm{Without}\:\infty} {\underbrace{\mathrm{5}×\mathrm{55}×\mathrm{555}×...}}\:=\:{t}\:\Rightarrow\:{t}\:>\:\mathrm{0} \\ $$$$\Rightarrow\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}{a}_{{n}} \:=\:{t}×\infty\:=\:\infty \\ $$$$\forall{t}\:\left\{{t}\:\in\:\mathbb{R}\:\mid\:{t}×\infty\:=\:\infty\right\} \\ $$$$\Rightarrow\:\mathrm{5}×\mathrm{55}×\mathrm{555}×...×\infty\:=\:\underline{\infty} \\ $$ | ||