Question: Use the definition of a Taylor series to find the first four nonzero terms of the series for \(…

Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a \). (Enter your answer as a comma-separated list.)

\[ f(x) = 3xe^x, \quad a = 0 \]

Solution

To find the Taylor series of \( f(x) = 3x e^x \) centered at \( a = 0 \), follow these steps: \[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} & \] Multiply both sides by \( 3x \): \[ 3x e^x = 3x \sum_{n=0}^{\infty} \frac{x^n}{n!} = \sum_{n=0}^{\infty} \frac{3x^{n+1}}{n!} & \] Expand the first four terms of the series: For \( n = 0 \): \[ \frac{3x^{1}}{0!} = 3x & \] For \( n = 1 \): \[ \frac{3x^{2}}{1!} = 3x^2 & \] For \( n = 2 \): \[ \frac{3x^{3}}{2!} = \frac{3}{2}x^3 & \] For \( n = 3 \): \[ \frac{3x^{4}}{3!} = \frac{3}{6}x^4 = \frac{1}{2}x^4 & \] First four nonzero terms of the Taylor series: \( 3x \), \( 3x^2 \), \( \frac{3}{2}x^3 \), \( \frac{1}{2}x^4 \)