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Let $\left\{a_n\right\}_{n\ge1}$ denote the sequence of functions from $\mathbb{R}^+$ to $\mathbb{R}$ defined, for every $t>0$, by $a_1(t)=t$ and, for every $n\ge1$, $$a_{n+1}(t)=a_n(t)-1+e^{-a_n(t)}.$$ Let $g(t)=\sum\limits_{n=1}^{\infty}a_n(t)$. Find $$\lim\limits_{t\to\infty}\ \left|g(t)-\tfrac{t(t+1)}{2}\right|.$$

What I tried:

For every $t>0$, $\left\{a_n(t)\right\}_{n\ge1}$ is a decreasing sequence. And $a_{n+1}(t) = t-n+\sum\limits_{k=1}^{n}e^{-a_k(t)}<t$.

From there I got no idea.

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  • $\begingroup$ you know that $e^{x} = \sum_{k=0}^\infty \frac{x^k}{k!}$, it means that if $|x|$ is small enough then $|e^x-1-x| < x^2$ $\endgroup$ – reuns Jul 29 '16 at 0:52
  • $\begingroup$ This doesn't help. How do we show $g(t)$ is valid for all $t\in\mathbb{R^+}$? Clearly $a_n(t)=O(t)$. $\endgroup$ – JohnDoe Jul 29 '16 at 1:37
  • $\begingroup$ if $|a_{n+1}| < a_n^2$ then clearly $\sum_{n=1}^\infty a_n$ converges whenever $|a_N| < 1$ for some $N$ $\endgroup$ – reuns Jul 29 '16 at 1:38
  • $\begingroup$ Yeah I managed to show that $g(t)\le t[t]+k =O(t^2);\ k<1$ but from here I have no idea how to find the above limit. $\endgroup$ – JohnDoe Jul 30 '16 at 18:06
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    $\begingroup$ @user1952009 did you actually find the limit? $\endgroup$ – JohnDoe Jul 30 '16 at 18:24
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This is not a solution. I am providing a plausibility argument to show the limit is of the form $\dfrac{e}{(e-1)^2} + \delta$, where $\delta$ is ''small'' (but unfortunately not too small).

Note if $a_k(t) > 0$, we have $a_k(t) - a_{k+1}(t) = 1 - \exp(-a_k(t)) > 0$ moreover since $e^x > 1 + x $ for $ x \neq 0$ we also have $a_{k+1} > 0.$ This implies $a_n(t)$ is a strictly decreasing positive sequence which converges to $a(t)$ where $a(t) = a(t) - 1 + e^{-a(t)}$ so $a(t) = 0.$

Also,$\lim_{n\to\infty}\dfrac{a_{n+1}(t)}{a^2_n(t)} = \lim_{n\to\infty}\dfrac{a_n(t) - 1 + \exp(-a_n(t))}{a_n^2(t)} = \dfrac{1}{2}.$ This suggests $a_n(t)$ converges to $0$ rapidly.

Lets analyze $(a_n(k))$ (denoted below as $(a_n)$ for simplicity) where $k$ is a large positive integer. So we have: $$ \begin{align} a_1 &= k, \\ a_2 &= k - 1 + e^{-k},\\ a_3 &= k - 2 + e^{-k} + e^{-((k - 1) + e^{-k})}\\ &= k - 2 + e^{-k} + e^{-(k-1)} + e^{-(k-1)}({e^{-e^{-k}} - 1})\\ &\approx k - 2 + e^{-k} + e^{-(k-1)},\\ a_4 &\approx k - 3 + e^{-k} + e^{-(k-1)} + e^{-(k-2)},\\ \end{align} $$

Assuming the approximation to continue, and assuming the terms $a_{k+2},\dots$ make small contribution to the sum, we have the following approximation to the sum: $$ \begin{align} a_1 + a_2 + \dots + a_{k+1} &\approx k + (k-1) + \dots + 0 + ke^{-k} + (k-1)e^{-(k-1)} + \dots e^{-1}\\ & \approx \dfrac{k(k+1)}{2} + \sum_{k=1}^{\infty} k e^{-k}\\ & \approx \dfrac{k(k+1)}{2} + \dfrac{e}{(e-1)^2} = \dfrac{k(k+1)}{2} + .92067359420779231894. \end{align} $$

The above provides an approximation of the limit as $.92067359420779231894$. Using the multi-precision bc calculator I got an estimate of the true limit as $.941282470970473$, so the above approximation is within 2%.

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  • $\begingroup$ Nice. Numerically, do you find the same limit when $t$ is not an integer, for example for $t=k+\frac12$, $k$ integer, $k\to\infty$? $\endgroup$ – Did Jul 31 '16 at 22:42
  • $\begingroup$ I haven't tried. I will check. $\endgroup$ – Arin Chaudhuri Jul 31 '16 at 23:47
  • $\begingroup$ I do notice a small difference, but it may vanish in the limit. $\endgroup$ – Arin Chaudhuri Aug 1 '16 at 1:54

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