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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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Why is the following recurrent sequence convergent?

Let $a, b , c, d$ be reals. Define the sequence $(x_n)$ as: $$x_0 = a,\,\, x_1 = b$$ $$x_n = \left(1 - \frac{b^2}{n^2}\right)x_{n-1} + \frac{1}{n-1}\sum_{k=0}^{n-2}\binom{2n+1}{2k+1}^{-1} (x_{k+1}-x_k …
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