To find the Maclaurin series for $f(x) = \frac{1}{(1+x)^3}$, we can use the binomial series expansion. The general formula for the binomial series is:

$(1 + x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n$

where $\binom{k}{n}$ (the binomial coefficient) is defined as:

$\binom{k}{n} = \frac{k(k-1)(k-2)\cdots(k-n+1)}{n!}, \quad \text{for } n \geq 1, \quad \binom{k}{0} = 1.$

Here, $k = -3$, since the function is $(1 + x)^{-3}$. Substituting $k = -3$ into the binomial expansion formula, we get:

$\frac{1}{(1+x)^3} = (1+x)^{-3} = \sum_{n=0}^{\infty} \binom{-3}{n} x^n$

Step 1: Compute the binomial coefficient

For $\binom{-3}{n}$, the formula becomes: $\binom{-3}{n} = \frac{(-3)(-3-1)(-3-2)\cdots(-3-n+1)}{n!} = \frac{(-3)(-4)(-5)\cdots(-3-n+1)}{n!}.$

Step 2: Write the series

Let us explicitly write the first few terms of the series:

$\frac{1}{(1+x)^3} = \sum_{n=0}^{\infty} \binom{-3}{n} x^n.$

For $n = 0$: $\binom{-3}{0} = 1 \quad \Rightarrow \quad \text{term: } 1.$

For $n = 1$: $\binom{-3}{1} = \frac{-3}{1!} = -3 \quad \Rightarrow \quad \text{term: } -3x.$

For $n = 2$: $\binom{-3}{2} = \frac{-3(-4)}{2!} = \frac{12}{2} = 6 \quad \Rightarrow \quad \text{term: } 6x^2.$

For $n = 3$: $\binom{-3}{3} = \frac{-3(-4)(-5)}{3!} = \frac{-60}{6} = -10 \quad \Rightarrow \quad \text{term: } -10x^3.$

For $n = 4$: $\binom{-3}{4} = \frac{-3(-4)(-5)(-6)}{4!} = \frac{360}{24} = 15 \quad \Rightarrow \quad \text{term: } 15x^4.$

Step 3: Write the general form and the first few terms

Thus, the Maclaurin series for $\frac{1}{(1+x)^3}$ is:

$f(x) = \frac{1}{(1+x)^3} = \sum_{n=0}^{\infty} \binom{-3}{n} x^n = 1 - 3x + 6x^2 - 10x^3 + 15x^4 - \cdots$

Final Answer:

$f(x) = \sum_{n=0}^{\infty} \binom{-3}{n} x^n = \sum_{n=0}^{\infty} \frac{(-3)(-3-1)(-3-2)\cdots(-3-n+1)}{n!} x^n.$

Do you want to see more detailed steps or examples?

---

5 Related Questions:

1. How can we use the binomial series to approximate $(1+x)^k$ for small $x$?
2. What is the radius of convergence of the Maclaurin series for $\frac{1}{(1+x)^3}$?
3. How would this series change if the function were $\frac{1}{(1-x)^3}$?
4. What are the applications of binomial series in real-world problems?
5. How can we use the derivative of the series to verify the expansion?

Tip:

When using the binomial series, always confirm the conditions for convergence ($|x| < 1$ in this case) to ensure validity!