List of top Engineering Mathematics Questions on Convergence tests asked in GATE EC

Consider a frequency-modulated (FM) signal \[ f(t) = A_c \cos(2\pi f_c t + 3 \sin(2\pi f_1 t) + 4 \sin(6\pi f_1 t)), \] where \( A_c \) and \( f_c \) are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency \( f_1 \) is in Hz, and assume that \( f_c>100 f_1 \). The peak frequency deviation of the FM signal in Hz is _________.

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

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different colours?

Consider the following series: \[ \sum_{n=1}^{\infty} \frac{n^{d}}{c^{n}} \] For which of the following combinations of $c, d$ values does this series converge?