List of top Real Analysis Questions asked in GATE MA

Let \( f : [-\pi, \pi] \to \mathbb{R} \) be a continuous function such that \( f(x)>\frac{f(0)}{2} \) for \( |x|<\delta \), where \( 0<\delta<\pi \). Define \( P_{n,\delta}(x) = (1 + \cos x - \cos \delta)^n \), for \( n = 1, 2, 3, \dots \). Then, which of the following statements is TRUE?

P: Suppose that \[ \sum_{n=0}^{\infty} a_n x^n \text{ converges at } x = -3 \text{ and diverges at } x = 6. \text{ Then } \sum_{n=0}^{\infty} (-1)^n a_n \text{ converges.} \] Q: The interval of convergence of the series \[ \sum_{n=2}^{\infty} \frac{(-1)^n x^n}{4^n \log_e n} \text{ is } [-4, 4]. \] Which of the following statements is TRUE?

Let \[ f_n(x) = \frac{x^2}{x^2 + (1 - nx)^2}, \quad x \in [0, 1], \, n = 1, 2, 3, \dots. \] Then, which of the following statements is TRUE?

Let \( f: \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \to \mathbb{R} \) be given by \( f(x) = \frac{\pi}{2} + x - \tan^{-1}x \).
Consider the following statements:
P: \( |f(x) - f(y)| < |x - y| \text{ for all } x, y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \).
Q: \( f \) has a fixed point.
Then:

Consider the following statements: P: \( d_1(x,y) = \left| \log \left( \frac{x}{y} \right) \right| \) is a metric on \( (0, 1) \).
Q: \( d_2(x, y) = \begin{cases} |x| + |y|, & \text{if } x \neq y \\ 0, & \text{if } x = y \end{cases} \) is a metric on \( (0, 1) \). Then:

Let \( f_n: [0, 10] \to \mathbb{R} \) be given by \( f_n(x) = n x^3 e^{-n x} \) for \( n = 1, 2, 3, \dots \). Consider the following statements: P: \( (f_n) \) is equicontinuous on \( [0, 10] \).
Q: \( \sum_{n=1}^{\infty} f_n \) does NOT converge uniformly on \( [0, 10] \). Then:

For each \( x \in (0, 1) \), consider the decimal representation \( x = d_1 d_2 d_3 \cdots d_n \cdots \). Define \( f: [0, 1] \to \mathbb{R} \) by \( f(x) = 0 \) if \( x \) is rational, and \( f(x) = 18n \) if \( x \) is irrational, where \( n \) is the number of zeroes immediately after the decimal point up to the first nonzero digit in the decimal representation of \( x \). Then the Lebesgue integral \[ \int_0^1 f(x) \, dx = \(\underline{\hspace{1cm}}\). \]