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Calculating infinite sum of a series

Posted 8 years ago
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I was curious about whether or not it was possible to make a fraction whose decimal expansion is all integers consecutively, i.e., 0.123456789101112... to infinity. I ended up expressing this value as the infinite sum of a sequence:

Sum[(10^((1/9) (-9 10^n n + 10^n - 10)) ((10^n)^(9 10^(n - 1)) (-10^n + 100^n + 10) - 10 (-10^n + 100^n + 1)))/(10^n - 1)^2, {n, 1, Infinity}]

This sequence generates a decimal expansion which goes up to the integer 10^x-1.where x is the upper bound of the summation. When I enter this into wolfram it doesn't provide any insight other than that the sum converges. I was just wondering if there was a way to simplify this further into a function in terms of x and whether this value can be expressed as a fraction at all. Is it irrational? Any insight at all would be very helpful.

POSTED BY: Philip Mann
3 Replies

And of course there is a Wolfram Language Function for that:

N[ChampernowneNumber[10], 25]
(*0.1234567891011121314151617*)

Cheers,

M.

POSTED BY: Marco Thiel
Posted 8 years ago

Perhaps

Champernowne Constant

or this

Champernowne Constant

might provide a little information for you.

POSTED BY: Bill Simpson
Posted 8 years ago

Thank you! You're my hero.

POSTED BY: Philip Mann
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