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?