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Reference:- Foundations of Functional Analysis, S. Ponnusammy, $\alpha-science-2002$

Consider the sequence $\{.1, .101, .101001, \ldots\}$, prove that given sequence is Cauchy and converges to a irrational number. find the limit of the sequence.

I tried to find the generalised formula for the sequence pattern. I failed to do. $\left\{\frac{1}{10}, \frac{1}{10}+\frac{1}{10^3}, \frac{1}{10}+\frac{1}{10^3}+\frac{1}{10^6}, \ldots\right\}$. It doesn't form any geometric series. How do I prove the sequence is Cauchy without getting the rigourous expression for the $n$-th term of the given sequence? how to find the limit of the sequence?

Michael Hardy
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  • These three initial elements of the sequence do not give any information on how it continues. Are you sure this is the full problem statement? – M. Winter Oct 30 '17 at 15:14
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    The $1$'s seem to be (I would guess) at the $1,3,6,10,\ldots$th position (so the $\frac{n(n+1)}{2}$th position for $n=1,2,3,\ldots$). If so then we can write $x = \sum_{n=1}^\infty 10^{-\frac{n(n+1)}{2}}$. – Winther Oct 30 '17 at 15:15
  • yes. I will post the picture of the problem. –  Oct 30 '17 at 15:15
  • You can write a formula (for what the sequence is, I assume, supposed to be): $a_n=\sum_{i=1}^n 10^{\sum_{k=1}^i k}$, but that's hardly helpful. Just take the difference between two elements and look at its decimal expansion. The estimates to make are trivial, you don't need any deep understanding of the sequence beyond the fact that its decimal expansion stabilises. – tomasz Oct 30 '17 at 15:17
  • Not completely on topic, but this video might be interesting. – M. Winter Oct 30 '17 at 15:19
  • I once asked a similar question. Answers are very interesting... https://math.stackexchange.com/questions/778218/is-0-1010010001000010000010000001-ldots-transcendental – Raffaele Oct 30 '17 at 15:47
  • The limit should be $$\frac{1}{2} \left(\sqrt[8]{10} \vartheta _2\left(0,\frac{1}{\sqrt{10}}\right)-2\right)$$ – Raffaele Oct 30 '17 at 15:50
  • The {curly braces} should not be excluded from within MathJax: $$\left{\frac{1}{10}, \frac{1}{10}+\frac{1}{10^3}, \frac{1}{10}+\frac{1}{10^3}+\frac{1}{10^6}, \ldots\right}$$ See my edits to the question. – Michael Hardy Oct 30 '17 at 15:52
  • @Raffaele How does the limit come like that? How to prove? –  Oct 30 '17 at 16:35
  • Can you please explain? what is the name of the function $\nu_2$? –  Oct 30 '17 at 16:39
  • I got it with Mathematica. It's pretty advanced math (functions of several complex variables) and it is called Elliptic Theta function. Read more here https://en.wikipedia.org/wiki/Theta_function – Raffaele Oct 30 '17 at 17:29

2 Answers2

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You can prove it is Cauchy directly from the definition. If I give you an $\epsilon \gt 0$ you just need to find an $N$ such that no two terms beyond $N$ differ by more than $\epsilon$. All the changes after $k$ add up to less than $10^{-k}$

Ross Millikan
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Hint: If $a$ and $b$ are positive, smaller than $1$, and they agree up to $k$ decimal places, then $\lvert a-b\rvert<\ldots$

tomasz
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  • A similar sequence could be with folowing general term:

    $\frac{101 . (10^n +1) - 100}{10^{2n}}$ which hods true for 3rd and further terms. The limit of this is:

    $Lim_{n→∞ } \frac{101 . (10^n +1) - 100}{10^{2n}}=Lim_{n→∞ }\frac{101 . (1/10^n +1) - 100/10^n}{10^{n}}=0$

     – sirous Oct 30 '17 at 20:51
  • @sirous: Yes, the behaviour of a sequence on the first three terms determines nothing about limits (or anything at all about any terms after the first three). Yes, the question is somewhat ill-posed. But I think it is clear what is meant, and I think your comment is beside the point. – tomasz Oct 30 '17 at 22:06