We are given a sequence of continuous functions \( \{ f_n \} \) on the interval \([0, 1]\), where the functions are defined as follows:
We need to determine which of the provided statements about the sequence is/are true.
The recurrence relation given for \( f_{n+1}(x) \) can be interpreted as a form of Newton's method for finding the square root of \( x \), with \( f_n(x) \approx \sqrt{x} \).
Notice that the recurrence relation:
is a form of iterative method intended to converge to the square root of \( x \). Particularly, given the form, this resembles the Newton-Raphson method for solving \((f_n(x))^2 = x\). It is known that Newton's method converges quadratically under suitable conditions.
Hence, \( f_n(x) \to \sqrt{x} \) pointwise on \([0, 1]\).
To determine if the convergence is uniform, consider the form:
This eventually reduces the error in each step, indicating uniform convergence over compact intervals like \([0, 1]\). Thus, the sequence converges uniformly on \([0, 1]\).
Therefore, the option claiming uniform convergence is correct.
We need to show that the functions are bound by:
We begin with \( f_1(x) = \frac{x}{2} \) and observe:
The claim \( \sqrt{x} - f_n(x) > \frac{2\sqrt{x}}{2 + n\sqrt{x}} \) is incorrect. With uniform convergence, the error diminishes rapidly, contradicting this lower bound.
Therefore, the correct answers are:
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?
To find a real root of the equation \( x^3 + 4x^2 - 10 = 0 \) in the interval \( \left( 1, \frac{3}{2} \right) \) using the fixed-point iteration scheme, consider the following two statements:
Statement 1 S1: The iteration scheme \( x_{k+1} = \sqrt{\frac{10}{4 + x_k}}, \, k = 0, 1, 2, \ldots \) converges for any initial guess \( x_0 \in \left( 1, \frac{3}{2} \right) \).
Statement 2 S2: The iteration scheme \( x_{k+1} = \frac{1}{2} \sqrt{10 - x_k^3}, \, k = 0, 1, 2, \ldots \) diverges for some initial guess \( x_0 \in \left( 1, \frac{3}{2} \right) \).