GATE EC 2023 Set 1 Q3 — Complex Analysis

Consider the two series, $S_{A}$ and $S_{B}$, where $$\begin{array}{l} S_{A}=\sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}} \\ S_{B}=1+\frac{1}{2}+\frac{1}{8}+\frac{1}{16}+\frac{1}{64}+\frac{1}{128}+\frac{1}{512}+\cdots \end{array}$$ Which of the following statements is correct for the two given series?

Consider the following series: $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ $\sum_{n=1}^{\infty} \frac{1}{n!}$ Choose the correct option.

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle $C$ in the complex plane is/are TRUE?

​​​​​Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ is $\_\_\_\_\_\_$.

The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is

A simple closed path $C$ in the complex plane is shown in the figure. If $$\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$$ where $i = \sqrt{-1},$ then the value of $A$ is _________ (rounded off to two decimal places).