Demonstrate that sequence below is convergent and calculate its limit.

The sequence is: $$X_{n+1} = X_n + (2 − e^{X_n})\left(\dfrac{X_n − X_{n−1}}{e^{X_n} - e^{X_{n-1}}}\right)$$ $$X_0 = 0, X_1 = 1$$


closed as off-topic by choco_addicted, Paul Frost, Alex Provost, Rebellos, José Carlos Santos Dec 22 '18 at 16:35

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  • 4
    $\begingroup$ You should probably explain what have you tried and what it is you have trouble with. We aren't gonna do your homework for you. $\endgroup$ – Javier May 27 '13 at 20:08
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    $\begingroup$ Somebody will, sadly. $\endgroup$ – Mark McClure May 27 '13 at 20:09
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    $\begingroup$ Tabulate $X_n$ against $n$ and study the pattern and you'll get there. $\endgroup$ – Maazul May 27 '13 at 20:20
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    $\begingroup$ This exercise is a bit more annoying than the average recursive sequence exercise. Little hint: whatever $x_n$ and $x_{n-1}$ are (as long as they are distinct), the ratio $\frac{x_n-x_{n-1}}{e^{x_n}-e^{x_{n-1}}}$ is positive. So if $x_n<\ln 2$, $x_{n+1}>x_{n}$, and if $x_n>\ln 2$, then $x_{n+1}<x_{n}$. $\endgroup$ – Julien May 27 '13 at 20:59