<s>[INST] What is the $n$ times iterate of $f(x)=\frac{x}{\sqrt{1+x^2}}$? We were asked to determine  the composition $f \circ f \circ f \circ \cdots \circ f $,  $n$ times, where $f(x)=\dfrac{x}{\sqrt{1+x^2}}$.
Has anyone an idea?
 [/INST] $$f(x)=\frac{x}{\sqrt{1+x^2}}$$
we see that
$$f^2(x)=f(f(x))=\frac{\frac{x}{\sqrt{1+x^2}}}{\sqrt{1+(\frac{x}{\sqrt{1+x^2}})^2}}=\frac{x}{\sqrt{1+2x^2}}$$
...... continuing this way
$$f^n(x)=\frac{x}{\sqrt{1+nx^2}}$$
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