from sympy import symbols, sqrt, series, Function

Define the variable and function

x = symbols('x') f = (1 - sqrt(1 + x2)) / (1 + sqrt(1 + x2))

Find the Maclaurin series (expansion about x=0)

maclaurin_series = series(f, x, 0, 6) # Expanding to 6 terms maclaurin_series-x2/4 + x4/8 + O(x6)-x2/4 + x4/8 + O(x6)successThe Maclaurin series expansion of the given function $f(x) = \frac{1 - \sqrt{1 + x^2}}{1 + \sqrt{1 + x^2}}$ up to the $x^4$ term is:

$f(x) = -\frac{x^2}{4} + \frac{x^4}{8} + \mathcal{O}(x^6)$

Would you like a detailed explanation of how the series is derived, or do you have any specific questions about this result?

Here are five related questions that expand on this topic:

1. What is the significance of the Maclaurin series in approximating functions?
2. How does the behavior of this function near $x = 0$ compare with the series approximation?
3. What happens if you compute more terms in the Maclaurin series?
4. Can this series be used to find limits or analyze the behavior of the function?
5. How does this function relate to other common mathematical series?

Tip: The Maclaurin series is particularly useful for approximating complex functions near $x = 0$, making calculations simpler in applied contexts.