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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.

The problem set you uploaded is focused on binomial expansions and their applications in approximations, a common topic in calculus and algebra.

Explore binomial expansions in calculus with problems on √(4 - x), approximating √3, and expanding functions like x/(1 - x)^3. Learn key formulas and theorems for solving these types of problems. Suitable for college-level students in calculus courses.

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