Find the 1 st 3 terms of the \( (1-1 / 3 x)^{-3} \)

To find the first three terms of the binomial expansion of \( (1 - \frac{1}{3}x)^{-3} \), we can use the binomial series formula, which states: \[ (1 + u)^n = \sum_{k=0}^{\infty} \binom{n}{k} u^k \] In our case, \( u = -\frac{1}{3}x \) and \( n = -3 \). The first three terms of the expansion can be calculated as follows: 1. For \( k = 0 \): \[ \binom{-3}{0} \left(-\frac{1}{3}x\right)^0 = 1 \] 2. For \( k = 1 \): \[ \binom{-3}{1} \left(-\frac{1}{3}x\right)^1 = -3 \cdot \left(-\frac{1}{3}x\right) = x \] 3. For \( k = 2 \): \[ \binom{-3}{2} \left(-\frac{1}{3}x\right)^2 = \frac{(-3)(-4)}{2!} \cdot \left(\frac{1}{9}x^2\right) = 6 \cdot \frac{1}{9}x^2 = \frac{2}{3}x^2 \] Putting it all together, the first three terms of the expansion are: \[ 1 + x + \frac{2}{3}x^2 \]