True or False: If x ∉ Q then ∑ m ≥ 0 m x m − 1 ∉ Q ,, where | x | &lt; 1.Considered the contra-positive of the above statement: If ∑ m ≥ 0 m x m − 1 ∈ Q then x ∈ Q .Now if the contra-positive is true/false then the statement is true/false.Thus, if ∑ m ≥ 0 m x m − 1 ∈ Q ⟹ 1 + 2 x + 3 x 2 + . . . ∈ Q ⟹ 1 ′ + x ′ + ( x 2 ) ′ + ( x 3 ) ′ + . . . ∈ Q ⟹ ( 1 + x + x 2 + x 3 + . . . ) ′ ∈ Q ⟹ ( 1 1 − x ) ′ ∈ Q ⟹ − 1 ( 1 − x ) 2 ∈ Q Now from the last step we can conclude that x ∈ Q , since | x | &lt; 1. Therefore the contrapositive is true, so the given statement is also true.Is my analysis correct?

Question

True or False: If   x  ∉      Q   then       ∑          m      ≥      0        m      x          m      −      1        ∉      Q    ,, where       |    x      |    &amp;lt;  1.Considered the contra-positive of the above statement: If       ∑          m      ≥      0        m      x          m      −      1        ∈      Q   then   x  ∈      Q    .Now if the contra-positive is true/false then the statement is true/false.Thus, if      ∑          m      ≥      0        m      x          m      −      1        ∈      Q      ⟹    1  +  2  x  +  3      x    2    +  .  .  .  ∈      Q      ⟹        1    ′    +      x    ′    +  (      x    2        )    ′    +  (      x    3        )    ′    +  .  .  .  ∈      Q      ⟹    (  1  +  x  +      x    2    +      x    3    +  .  .  .      )    ′    ∈      Q      ⟹    (      1          1      −      x            )    ′    ∈      Q      ⟹              −      1              (      1      −      x              )        2              ∈      Q  Now from the last step we can conclude that   x  ∈      Q  , since       |    x      |    &amp;lt;  1. Therefore the contrapositive is true, so the given statement is also true.Is my analysis correct?

Ostrakodec3 · Accepted Answer

Take   x  =      2    −  1, you have             1              (        1        −        x                  )          2                      ∈      Q  .

True or False: If x∉Q then Sum m>=0mxm−1∉Q,, where |x|<1. Considered the contra-positive of the above statement: If Sum m>=0mxm−1∈Q then x∈Q. Now if the contra-positive is true/false then the statement is true/false.

Answered question

Answer & Explanation

New Questions in College Statistics