Monday, 20 August 2018

sequences and series - Compute: limlimitsntoinftyfracxnlnn



Let be the sequence (xn)n0, x0 a real number, and defined as follows: xn+1=xn+exn
Compute the limit:



lim




Luckily, I've found another post here with a very similar question. If you know other ways to solve it and want to share them then I'd be very grateful for that.


Answer



Heuristically, this difference equation resembles the differential equation x'(t) = e^{-x}, which has solutions x(t) = \ln(t + C), so I would expect the answer to be 1.



In fact, let x_n = \ln(n + s_n), and y(t) = \ln(t+s_n). Then for n \le t < n+1 we have
y(t) > y(n) = x_n, so
\eqalign{ \ln(n+1+s_n) - \ln(n+s_n) &= \int_n^{n+1} \dfrac{dt}{t+s_n} = \int_n^{n+1} e^{-y(t)}\ dt\cr &< \int_n^{n+1} e^{-x_n}\ dt = e^{-x_n} = x_{n+1} - x_n\cr}

i.e. x_{n+1} - \ln(n+1+s_n) > x_n - \ln(n+s_n) = 0, or s_{n+1} > s_n.
Thus x_n > \ln(n+s_0). But then x_{n+1} - x_n = e^{-x_n} < \dfrac{1}{n + s_0}
so \displaystyle x_n < x_0 + \sum_{j=0}^{n-1} \frac{1}{j+s_0}. Since both upper and lower bounds are \ln(n) + O(1), we conclude that x_n = \ln(n) + O(1).


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