1
GATE ECE 2022
MCQ (More than One Correct Answer)
+1
-0.33

Consider the following series:

$$\sum\limits_{n = 1}^\infty {{{{n^d}} \over {{c^n}}}}$$

For which of the following combinations of c, d values does this series converge?

A
c = 1, d = $$-$$1
B
c = 2, d = 1
C
c = 0.5, d = $$-$$10
D
c = 1, d = $$-$$2
2
GATE ECE 2017 Set 2
+1
-0.3
The residues of a function $$f\left( z \right) = {1 \over {\left( {z - 4} \right){{\left( {z + 1} \right)}^3}}}$$ are
A
$${{ - 1} \over {27}}$$ and $${{ - 1} \over {125}}$$
B
$${1 \over {125}}$$ and $${{ - 1} \over {125}}$$
C
$${{ - 1} \over {27}}$$ and $${1 \over 5}$$
D
$${1 \over {125}}$$ and $${{ - 1} \over 5}$$
3
GATE ECE 2016 Set 3
Numerical
+1
-0
For $$f\left( z \right) = {{\sin \left( z \right)} \over {{z^2}}},$$ the residue of the pole at $$z=0$$ ________.
4
GATE ECE 2014 Set 2
+1
-0.3
The real part of an analytic function $$f(z)$$ where $$z=x+jy$$ is given by $${e^{ - y}}\cos \left( x \right).$$ The imaginary part of $$f(z)$$ is
A
$${e^y}\cos \left( x \right)$$
B
$${e^{ - y}}sin\left( x \right)$$
C
$$- {e^y}sin\left( x \right)$$
D
$$- {e^{ - y}}sin\left( x \right)$$
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