GATE XE 2026 | Question: 8

Tue, 24 Feb 2026 15:50:12 +0000

Match each entry of $\text{List-1}$ with a suitable entry in $\text{List-2}$ and choose the correct option.

$\textbf{List-1}$		$\textbf{List-2}$
$\mathrm{P}$	$\text{The sum of the series } \displaystyle \sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)} \text{ is equal to}$	$\text{I}$	$\dfrac{3}{2}$
$\mathrm{Q}$	$\displaystyle \lim_{x \to 0} \left( \frac{3}{x^2} \int_0^x \sin(t)\,dt \right) \text{ is equal to}$	$\text{II}$	$1$
$\mathrm{R}$	$ \text{Let } \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) $ $\text{ be the Fourier series expansion of} \text{the function } $ $f(x) = \dfrac{1}{2}\sin x - \dfrac{1}{2}\cos x + \dfrac{1}{\sqrt{2}}\sin 2x, $ $\ x \in [0,2\pi] \text{Then } \displaystyle \sum_{n=0}^{\infty} (a_n^2 + b_n^2) $ $\text{ is equal to}$	$\text{III}$	$\dfrac{1}{2}$

$\mathrm{P} \rightarrow \mathrm{I}, \mathrm{Q} \rightarrow \mathrm{III}, \mathrm{R} \rightarrow \mathrm{II}$
$\mathrm{P} \rightarrow \mathrm{III}, \mathrm{Q} \rightarrow \mathrm{II}, \mathrm{R} \rightarrow \mathrm{I}$
$\mathrm{P} \rightarrow \mathrm{III}, \mathrm{Q} \rightarrow \mathrm{I}, \mathrm{R} \rightarrow \mathrm{II}$
$\mathrm{P} \rightarrow \mathrm{II}, \mathrm{Q} \rightarrow \mathrm{I}, \mathrm{R} \rightarrow \mathrm{III}$

GATE XE 2025 | Question: 13

Sun, 04 May 2025 19:08:41 +0000

Consider the infinite series

$$ \begin{array}{l} \text { (P) }: \sum_{n=2}^{\infty} \frac{1}{(n \log n)^{1 / n}} \\
\text { (Q): } \sum_{n=1}^{\infty} \frac{n^{n}}{(2 n)!}
\end{array}$$

Then which one of the following statements is correct?

Series $(P)$ and $(Q)$ both converge
Series $(P)$ converges and series $(Q)$ diverges
Series $(P)$ and $(Q)$ both diverge
Series $(P)$ diverges and series $(Q)$ converges

GATE Overflow for GATE XE - Recent questions and answers in Sequences and Series

GATE XE 2026 | Question: 8

GATE XE 2025 | Question: 13