# Prove convergence of sequence and calculate its limit

Demonstrate that sequence below is convergent and calculate its limit.

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

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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. –  Javier Badia May 27 '13 at 20:08
Somebody will, sadly. –  Mark McClure May 27 '13 at 20:09
Tabulate $X_n$ against $n$ and study the pattern and you'll get there. –  Maazul May 27 '13 at 20:20
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 $x_{n+1}<x_{n}$. –  1015 May 27 '13 at 20:59