The series (z-1)-1-1+(z-1)-(z-1)2+ represents .

GATE 2021 IN – Question 36

Engineering Mathematics · Complex variables · 2 marks · Multiple choice

The series $(z-1)^{-1}-1+(z-1)-(z-1)^2+\cdots$ represents ______.

$-1/[z(z-1)]$, $|z-1|<1$
$1/[z(z-1)]$, $|z-1|<1$
$1/(z-1)^2$, $|z-1|<1$
$-1/(z-1)$, $|z-1|<1$

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Correct answer: (B) $1/[z(z-1)]$, $|z-1|<1$

Explanation

Set w=z−1. The series is 1/w−1+w−w²+⋯=(1/w)(1−w+w²−⋯)=1/[w(1+w)]. Since 1+w=z, it equals **1/[z(z−1)], B**, convergent for 0<|z−1|<1. The centre is excluded by the pole.