Prove that 2.101001000100001... is an irrational number.

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Trey Ross · Accepted Answer

Let   x  =  2  +      ∑          k      =      1        ∞        10          −      k      (      k      +      1      )              /            2       be the number at hand. If   x is rational, say  x  =      p    q   for some positive integers   p  ,  q, we can pick a   n  &amp;gt;  1 such that       10    n    &amp;gt;  q  +  1. It is clear  q  x  ×      10          n      (      n      −      1      )              /            2        =  p  ×      10          n      (      n      −      1      )              /            2      Is also an integer. However, the fractional part of this number is equal to      {    q    ×          10              n        (        n        −        1        )                  /                2                    (      2      +              ∑                  k          =          1                ∞                    10                  −          k          (          k          +          1          )                      /                    2                    )        }    =      {    q    ×          ∑              k        =        1            ∞              10              −        k        (        k        +        2        n        −        1        )                  /                2              }  Which belongs to   (  q  ×      10          −      n        ,  (  q  +  1  )  ×      10          −      n        )  ⊂  (  0  ,  1  ). Since   (  0  ,  1  ) doesn&#039;t contain any integer, this leads to a contradiction and hence   x is irrational.

Prove that 2.101001000100001... is an irrational number.

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