Residue of exp over power

• A.Wrong

  False.

  $I_{k} = 0$ only for k ≤ 0, where eᶻ/zᵏ is entire and Cauchy's theorem applies.

  For k ≥ 1 there is a pole at the origin: the residue is the coefficient of $z^{k-1}$ in the expansion of eᶻ, namely 1/(k−1)!

  so $I_{k} = 2\pi{}i/(k-1)! \ne 0$.

  SymPy gives |$I_{1}$| $= 2\pi$.

• B.Correct

  True.

  $I_{k} = 2\pi{}i \cdot$ $Res_{z=0}(e$ᶻ$/z^{k}) = 2\pi{}i \cdot [$coefficient of $z^{k-1}$ in eᶻ$] = 2\pi{}i/(k-1)$!

  which is never zero for k ≥ 1 since factorials are finite.

  The trigger: for eᶻ/zᵏ the residue is just one Taylor coefficient — read it off rather than differentiating k − 1 times.

• C.WrongBoundary and endpoint

  False.

  The moduli are $2\pi/(k-1)$!

  which decrease once the factorial starts biting.

  SymPy gives |$I_{2}$| $= 2\pi$ and |$I_{3}$| $= \pi$

  so |$I_{2}$| > |$I_{3}$| and the claimed inequality fails at k = 2.

  The first two values being equal (|$I_{1}$| = |$I_{2}$| $= 2\pi$

  since 0! = 1! = 1) is what makes the sequence look non-decreasing at a glance.

• D.WrongExecution slip

  False — the moduli tend to 0, not infinity.

  SymPy confirms $\lim 2\pi/\Gamma(k) = 0$ as $k \to \infty$: the factorial in the denominator dominates everything.

  The values run $2\pi, 2\pi, \pi, \pi/3$, and shrink from there.

2 of these 3 wrong options correspond to a named reasoning error — the same errors recur across subjects.