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Please give me some information about decimal expansion of numbers so that I could try out this problem.

Rajath Radhakrishnan
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The sequence starts $0,\ 2,\ 4,\ 6,\ 8,\ 20,\ 22\ldots$.

Hint: The sequence contains $5$ numbers less than $10$, $25$ numbers less than $100$, and how many numbers less than $10^n$?

Servaes
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  • Please give me some information on decimal expansion before any hints on starting off with the answer. – Rajath Radhakrishnan Aug 26 '13 at 13:18
  • What would you like to know about decimal expansion? – Servaes Aug 26 '13 at 13:33
  • $25$ is the decimal expansion of twenty-five. $5^2$ is not. I would have preferred "decimal expression" or "usual expression" – Henry Aug 26 '13 at 13:35
  • @Servaes I would like to know about writing the decimal expansion of a number. What type of restriction rise up whwn it is given that the decimal expansion should only include 0,2,4,6,8. – Rajath Radhakrishnan Aug 26 '13 at 13:38
  • Precisely the restriction that is given; that the decimal expansion should only include $0,2,4,6,8$. For example, the number $8642$ has only even digits, whereas $8641$ does not. This is all you need to know to solve the problem. – Servaes Aug 26 '13 at 14:52
  • But how to find the nth term in terms of n? After that I could substitute for it in the given equation and use L-Hospital rule to find the limit. – Rajath Radhakrishnan Aug 27 '13 at 13:03
  • The $n$-th term of the sequence is the quinary expansion of $n$ viewed as a decimal expansion, multiplied by two. As far as I know it is not (easily) expressable in a way that extends to a differentiable function on $\Bbb{R}_{>0}$. I would advise against trying to use L'Hôpital's rule. In stead, try using the squeeze theorem. – Servaes Aug 27 '13 at 13:08