We need to expand the two given functions using the binomial series expansion up to the $x^3$ term and also determine the radius of convergence for each series.

1. Expansion of $(1 + x^2)^{-1}$:

We use the binomial expansion formula for $(1 + u)^n$, which is given by:

$(1 + u)^n = 1 + nu + \frac{n(n-1)}{2!}u^2 + \frac{n(n-1)(n-2)}{3!}u^3 + \cdots$

For $(1 + x^2)^{-1}$, we have $u = x^2$ and $n = -1$. Substituting these values into the binomial expansion:

$(1 + x^2)^{-1} = 1 + (-1)x^2 + \frac{(-1)(-2)}{2!}x^4 + \frac{(-1)(-2)(-3)}{3!}x^6 + \cdots$

Now, let's expand up to and including the $x^3$ term:

$(1 + x^2)^{-1} = 1 - x^2 + \cdots$

Thus, the expansion of $(1 + x^2)^{-1}$ up to the $x^3$ term is:

$1 - x^2 + O(x^4)$

Radius of Convergence for $(1 + x^2)^{-1}$:

The binomial series $(1 + u)^n$ converges when $|u| < 1$. In this case, $u = x^2$, so the series converges when:

$|x^2| < 1 \quad \Rightarrow \quad |x| < 1$

Thus, the series for $(1 + x^2)^{-1}$ converges for $|x| < 1$.

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2. Expansion of $(1 - x)^{-1}$:

Now we consider the binomial expansion for $(1 - x)^{-1}$. Using the same binomial expansion formula with $u = -x$ and $n = -1$, we get:

$(1 - x)^{-1} = 1 + (-1)(-x) + \frac{(-1)(-2)}{2!}(-x)^2 + \frac{(-1)(-2)(-3)}{3!}(-x)^3 + \cdots$

Expanding up to and including the $x^3$ term:

$(1 - x)^{-1} = 1 + x + \frac{2}{2}x^2 + \frac{6}{6}x^3 + O(x^4)$

Thus, the expansion of $(1 - x)^{-1}$ up to the $x^3$ term is:

$1 + x + x^2 + x^3 + O(x^4)$

Radius of Convergence for $(1 - x)^{-1}$:

For $(1 - x)^{-1}$, the series converges when $|x| < 1$, since this is a standard geometric series with the form $\frac{1}{1 - x}$, which converges for $|x| < 1$.

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Summary:

1. The expansion of $(1 + x^2)^{-1}$ up to $x^3$ is:

$1 - x^2 + O(x^4)$

The series converges for $|x| < 1$.

2. The expansion of $(1 - x)^{-1}$ up to $x^3$ is:

$1 + x + x^2 + x^3 + O(x^4)$

The series converges for $|x| < 1$.

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Would you like to explore more about binomial expansions or the convergence of series? Here are some related questions:

1. How does the binomial expansion work for non-integer exponents?
2. What is the general form of the binomial series for arbitrary $n$?
3. How can the binomial expansion be applied to functions other than polynomials, such as trigonometric or exponential functions?
4. What happens when the expansion is carried out beyond the $x^3$ term?
5. How do we apply the binomial expansion to functions of the form $(1 + ax^k)^n$?

Tip: The radius of convergence of a binomial series expansion is crucial in determining where the approximation holds true; outside this radius, the series may not provide accurate results.