The problem set you uploaded is focused on binomial expansions and their applications in approximations, a common topic in calculus and algebra. Here’s a breakdown of the first few problems:

1. Binomial expansion of $\sqrt{4 - x}$:

   • You are tasked with expanding $\sqrt{4 - x}$ using the binomial theorem and finding the coefficient $k$. You will need to use the form: $(1 + u)^n = 1 + nu + \frac{n(n-1)}{2!}u^2 + \dots$ where $u = -\frac{x}{4}$ and $n = \frac{1}{2}$.

2. Binomial expansion for $\frac{\sqrt{1+x}}{\sqrt{1-x}}$:

   • This requires approximating using the binomial theorem and substituting $x = \frac{1}{26}$ to estimate $\sqrt{3}$.

3. Expansion of $\frac{x}{(1 - x)^3}$:

   • You are asked to find the expansion of the given function in powers of $x$. This involves using the binomial series and simplifying the result.

Would you like assistance with solving any specific problem from this set?

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Here are 5 related questions to expand the discussion:

1. How do you apply the binomial theorem to functions of the form $(1 - x)^n$?
2. What is the range of $x$ for which binomial expansions are valid for these problems?
3. Can you use binomial expansions to approximate values for large $x$, or only small $x$?
4. How does substituting specific values of $x$ impact the accuracy of the approximation?
5. What are other uses of binomial expansions in calculus beyond approximations?

Tip: When performing binomial expansions for roots like $\sqrt{4 - x}$, always start by factoring constants for easier computation.