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CSS Pure Mathematics QUESTION #1272
Question 1
The Maclaurin series for \(f(Z)=Z^2e^{3Z}\) starts with:
  • \(Z^2+3Z^3+\dfrac{9}{2}Z^4+\cdots\)✔️
  • \(Z^2+3Z^3+3Z^4+\cdots\)
  • \(Z^2+Z^3+\dfrac{1}{2}Z^4+\cdots\)
  • \(1+Z^2+3Z^3+\cdots\)
Correct Answer Explanation
Use \(e^{3Z}=\sum_{n=0}^\infty\dfrac{(3Z)^n}{n!}=1+3Z+\dfrac{9Z^2}{2}+\dfrac{9Z^3}{2}+\cdots\). Multiply by \(Z^2\): \(Z^2e^{3Z}=Z^2+3Z^3+\dfrac{9Z^4}{2}+\cdots\).