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GATE 2023 IN – Question 19

Engineering Mathematics · Complex variables · 1 mark · Multiple choice

Expanding $F(z)=1/(1-z)$ around $z=2$ gives $F(z)=\sum_{k=0}^\infty a_k(z-2)^k$, with ROC $|z-2|<1$. For $k\ge0$, the coefficients are

  1. $(-1)^k$
  2. $(-1)^{k+1}$
  3. $(1/2)^k$
  4. $(-1/2)^{k+1}$

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Show answer and explanation

Correct answer: (B) $(-1)^{k+1}$

Explanation

Let $w=z-2$. Then $1/(1-z)=-1/(1+w)=-\sum_{k=0}^\infty(-w)^k$ for $|w|<1$. The coefficient is $a_k=(-1)^{k+1}$, **B**, and the ROC is $|z-2|<1$.