Can we say that&nbsp;2&nbsp;=&nbsp;2&nbsp;/&nbsp;(&nbsp;2&nbsp;/&nbsp;(&nbsp;2&nbsp;/&nbsp;(&nbsp;…&nbsp;)&nbsp;)&nbsp;)?We haveϕ&nbsp;=&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;ϕ&nbsp;=&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;ϕ&nbsp;=&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;1&nbsp;ϕ&nbsp;=&nbsp;⋯(with&nbsp;ϕ&nbsp;being the Golden Ratio)It provides the verified infinite fraction.ϕ&nbsp;=&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;⋱&nbsp;We also have2&nbsp;=&nbsp;&nbsp;2&nbsp;&nbsp;2&nbsp;=&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;2&nbsp;=&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;2&nbsp;=&nbsp;⋯Hence, we might conclude that2&nbsp;=&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;&nbsp;2&nbsp;&nbsp;⋮&nbsp;The sequence&nbsp;(&nbsp;a&nbsp;n&nbsp;)&nbsp;n&nbsp;∈&nbsp;Z&nbsp;+&nbsp;such that&nbsp;a&nbsp;1&nbsp;=&nbsp;2&nbsp;,&nbsp;a&nbsp;n&nbsp;+&nbsp;1&nbsp;=&nbsp;2&nbsp;a&nbsp;n&nbsp;gives&nbsp;lim&nbsp;n&nbsp;→&nbsp;+&nbsp;∞&nbsp;a&nbsp;n&nbsp;=&nbsp;2&nbsp;, so indeed the representation should be correct.Wherever I look on the Internet, there is a constant&nbsp;2&nbsp;, it is&nbsp;2&nbsp;=&nbsp;1&nbsp;+&nbsp;(&nbsp;2&nbsp;−&nbsp;1&nbsp;)&nbsp;=&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;2&nbsp;=&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;2&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;2&nbsp;=&nbsp;1&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;2&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;2&nbsp;+&nbsp;&nbsp;1&nbsp;&nbsp;1&nbsp;+&nbsp;2&nbsp;=&nbsp;⋯Why haven't I seen the representation&nbsp;2&nbsp;=&nbsp;2&nbsp;/&nbsp;(&nbsp;2&nbsp;/&nbsp;(&nbsp;2&nbsp;/&nbsp;(&nbsp;2&nbsp;/&nbsp;…&nbsp;)&nbsp;)&nbsp;)&nbsp;mentioned, and see the above used instead? Is something wrong about my representation?It is not stated since, in my opinion, it is not helpful: you cannot approximate&nbsp;2&nbsp;utilizing the representation as opposed to the one above (which may be removed),&nbsp;1&nbsp;1&nbsp;+&nbsp;2&nbsp;n any member).

Question

Can we say that&amp;nbsp;2&amp;nbsp;=&amp;nbsp;2&amp;nbsp;/&amp;nbsp;(&amp;nbsp;2&amp;nbsp;/&amp;nbsp;(&amp;nbsp;2&amp;nbsp;/&amp;nbsp;(&amp;nbsp;…&amp;nbsp;)&amp;nbsp;)&amp;nbsp;)?We haveϕ&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;ϕ&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;ϕ&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;1&amp;nbsp;ϕ&amp;nbsp;=&amp;nbsp;⋯(with&amp;nbsp;ϕ&amp;nbsp;being the Golden Ratio)It provides the verified infinite fraction.ϕ&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;⋱&amp;nbsp;We also have2&amp;nbsp;=&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;2&amp;nbsp;=&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;2&amp;nbsp;=&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;2&amp;nbsp;=&amp;nbsp;⋯Hence, we might conclude that2&amp;nbsp;=&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;&amp;nbsp;2&amp;nbsp;&amp;nbsp;⋮&amp;nbsp;The sequence&amp;nbsp;(&amp;nbsp;a&amp;nbsp;n&amp;nbsp;)&amp;nbsp;n&amp;nbsp;∈&amp;nbsp;Z&amp;nbsp;+&amp;nbsp;such that&amp;nbsp;a&amp;nbsp;1&amp;nbsp;=&amp;nbsp;2&amp;nbsp;,&amp;nbsp;a&amp;nbsp;n&amp;nbsp;+&amp;nbsp;1&amp;nbsp;=&amp;nbsp;2&amp;nbsp;a&amp;nbsp;n&amp;nbsp;gives&amp;nbsp;lim&amp;nbsp;n&amp;nbsp;→&amp;nbsp;+&amp;nbsp;∞&amp;nbsp;a&amp;nbsp;n&amp;nbsp;=&amp;nbsp;2&amp;nbsp;, so indeed the representation should be correct.Wherever I look on the Internet, there is a constant&amp;nbsp;2&amp;nbsp;, it is&amp;nbsp;2&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;(&amp;nbsp;2&amp;nbsp;−&amp;nbsp;1&amp;nbsp;)&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;2&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;2&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;2&amp;nbsp;=&amp;nbsp;1&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;2&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;2&amp;nbsp;+&amp;nbsp;&amp;nbsp;1&amp;nbsp;&amp;nbsp;1&amp;nbsp;+&amp;nbsp;2&amp;nbsp;=&amp;nbsp;⋯Why haven&#039;t I seen the representation&amp;nbsp;2&amp;nbsp;=&amp;nbsp;2&amp;nbsp;/&amp;nbsp;(&amp;nbsp;2&amp;nbsp;/&amp;nbsp;(&amp;nbsp;2&amp;nbsp;/&amp;nbsp;(&amp;nbsp;2&amp;nbsp;/&amp;nbsp;…&amp;nbsp;)&amp;nbsp;)&amp;nbsp;)&amp;nbsp;mentioned, and see the above used instead? Is something wrong about my representation?It is not stated since, in my opinion, it is not helpful: you cannot approximate&amp;nbsp;2&amp;nbsp;utilizing the representation as opposed to the one above (which may be removed),&amp;nbsp;1&amp;nbsp;1&amp;nbsp;+&amp;nbsp;2&amp;nbsp;n any member).

Kaylie Mcdonald · Accepted Answer

You have:&amp;nbsp;2&amp;nbsp;:=&amp;nbsp;2&amp;nbsp;2&amp;nbsp;2&amp;nbsp;2&amp;nbsp;⋮&amp;nbsp;2&amp;nbsp;=&amp;nbsp;2&amp;nbsp;2&amp;nbsp;⋅&amp;nbsp;2&amp;nbsp;2&amp;nbsp;⋯&amp;nbsp;2&amp;nbsp;That definition is not very useful.

Salvador Bush · Accepted Answer

Cauchy sequences are used to generate&amp;nbsp;R&amp;nbsp;The sequence&#039;s constraint&amp;nbsp;{&amp;nbsp;x&amp;nbsp;i&amp;nbsp;}&amp;nbsp;where&amp;nbsp;x&amp;nbsp;1&amp;nbsp;=&amp;nbsp;2,&amp;nbsp;x&amp;nbsp;n&amp;nbsp;=&amp;nbsp;2&amp;nbsp;x&amp;nbsp;n&amp;nbsp;−&amp;nbsp;1&amp;nbsp;does not exist. Thus&amp;nbsp;{&amp;nbsp;x&amp;nbsp;i&amp;nbsp;}&amp;nbsp;not a cauchy sequence.

Can we say that <msqrt> 2 </msqrt> = 2 <mrow class="MJX-TeXAtom-ORD"> /

Answered question

Answer & Explanation

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