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Question Number 4242 by prakash jain last updated on 05/Jan/16

$$\mathrm{Find}\mathrm{that}\mathrm{value}\mathrm{of}$$$$2^{2^{2\cdot \cdot \cdot }}\left(\mathrm{continued}\mathrm{power}\mathrm{of}2\right)$$$$\mathrm{using}\mathrm{analytical}\mathrm{continuation}.$$

Commented by RasheedSindhi last updated on 06/Jan/16

$$S=2^{2^{2\cdot \cdot \cdot }}\Rightarrow S=2^S\Rightarrow {\log }_{2}S=S{\log }_{2}2$$$${\log }_{2}S=S$$$$$$$$.....$$

Commented by Rasheed Soomro last updated on 06/Jan/16

$$S=2^{2^{2\cdot \cdot \cdot }}\Rightarrow S=2^S\Rightarrow S^{1/S}=2$$$$\mathrm{Graph}\mathrm{of}{\mathrm{S}}^{1/\mathrm{S}}showsthat{\mathrm{S}}^{1/\mathrm{S}}<2\mathrm{for}\mathrm{always}.$$$$\mathrm{So}\mathrm{there}\mathrm{is}\mathrm{no}\mathrm{real}\mathrm{S}.$$$$\mathrm{See}\mathrm{the}\mathrm{graph}\mathrm{below}\mathrm{in}\mathrm{next}\mathrm{comment}.$$

Commented by Filup last updated on 07/Jan/16

$$‘‘\mathrm{There}\mathrm{is}\mathrm{no}\mathrm{real}S{}^{″}$$$$$$$$\mathrm{So},2^{2^{2...}}\notin \mathbb{R}?$$

Commented by Rasheed Soomro last updated on 06/Jan/16

Commented by Yozzii last updated on 07/Jan/16

$$Ithinkwhatthismeansisthat{2}^{2^{2...}}{doesn}^{\prime }t$$$$convergeinvalue\left(obviously\right).$$$$SostatingthatS=2^{S}assumesthe$$$$conergenceof{2}^{2^{2..}}toavaluewhich$$$$wecalledS.Ofcourse,2^{2^{2^{...}}}\in \mathbb{R}since2\in \mathbb{R}.$$$$$$

Commented by Rasheed Soomro last updated on 07/Jan/16

$$Ithink{2}^{2^{2...}}{isn}^{\prime }tanumber.Itis\mathrm{\infty }which$$$$isnotanumber.Itisinthiscasealimit.$$$$Itispartofextendedrealnumbers.$$$$2^{2^{2...}}\notin \mathbb{R}$$

Commented by Yozzii last updated on 07/Jan/16

$$Whatabout{\left(\sqrt{2}\right)}^{{\left(\sqrt{2}\right)}^{{\left(\sqrt{2}\right)}^{{\left(\sqrt{2}\right)}^{...}}}}?Assuming$$$$convergenceofthisexpressionto$$$$sweget,s={\left(\sqrt{2}\right)}^s.Belowisagraph$$$$indicatingthats=2ors=4.$$$$$$

Commented by Yozzii last updated on 07/Jan/16

Commented by Yozzii last updated on 07/Jan/16

Commented by prakash jain last updated on 07/Jan/16

$$\mathrm{My}\mathrm{question}\mathrm{was}\mathrm{can}\mathrm{we}\mathrm{give}\mathrm{this}\mathrm{a}\mathrm{value}$$$$\mathrm{using}\mathrm{analytical}\mathrm{continuity}.$$$$\mathrm{For}\mathrm{example}$$$$1+2+2^2+2^4+..=-1$$

Commented by 123456 last updated on 08/Jan/16

$$a=0,999999....‘‘1^{″}$$$$b=...999‘‘-1^{″}$$$$c=...999,999....‘‘0^{″}$$

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