# Simplified Formula for Champernowne's constant

Recently I heard about Champernowne's constant which for those of you who have not heard of it, is the constant of the form .123456789101121314151617181920... including every possible string of digits. After reading about it I read this particularly nasty infinite series formula for this constant. I tried to create a more simplified formula for it and I came up with $$\sum _{n=1}^{\infty }\:\frac{n}{\left(10^{\sum _{k=1}^nfloor\left(log _{10}\left(k\right)\right)+1\:}\right)}$$ I have tried using several sites such as wolfram alpha and others to try to confirm my result but none would work correctly. Is there any way I can confirm that this formula truly is equivalent to Champernowne's constant? I would also be fine with using a program to iterate partial sums to see approximately what it converges to. Thanks!

• Computer tests show it converging to $0.120304050607008$ – Cheerful Parsnip May 15 '17 at 1:41
• it previously said the natural log and I have fixed it to be the common log or log base 10 which is what I originally meant. – Brothersquid May 15 '17 at 2:20
• Aha. That makes more sense, and computer experiments point toward it being correct. – Cheerful Parsnip May 15 '17 at 2:26