Let
![f(x) = x^n](/media/m/8/2/5/825c04def867dcb2749781e206d0c3ce.png)
where
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
is a fixed positive integer and
![x =1, 2, \cdots .](/media/m/7/0/a/70a7e3263e934369260dbbc73fc5b46c.png)
Is the decimal expansion
![a = 0.f (1)f(2)f(3) . . .](/media/m/7/0/9/7091bc759f79dcdfa10a862060438155.png)
rational for any value of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
?
The decimal expansion of a is defined as follows: If
![f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)](/media/m/2/f/3/2f38a3874bedda1e789f68e4a2fe9a40.png)
is the decimal expansion of
![f(x)](/media/m/3/f/4/3f40d68090aa4fb60a440be4675c7aca.png)
, then
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Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ?
The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$