List of top Real Analysis Questions asked in IIT JAM MA

Let $ f $ be a real-valued function of a real variable, such that $ |f^{(n)}(0)| \leq K $ for all $ n \in \mathbb{N} $, where $ K > 0. $ Which of the following is/are true?

If $ s_n = \dfrac{(-1)^n}{2^n + 3} $ and $ t_n = \dfrac{(-1)^n}{4n - 1}, \, n = 0, 1, 2, \dots, $ then

The sum of the series \[ \frac{1}{2(2^2 - 1)} + \frac{1}{3(3^2 - 1)} + \frac{1}{4(4^2 - 1)} + \cdots \] is ...........

Let $ x_n = n^{1/n} $ and $ y_n = e^{1 - x_n}, \, n \in \mathbb{N}. $ Then the value of $ \lim_{n \to \infty} y_n $ is ..............

For $ x \in \mathbb{R} $, let $ f(x) = \begin{cases} x^3 \sin \left( \frac{1}{x} \right), & x \neq 0 \\ 0, & x = 0 \end{cases} $ . Then which one of the following is FALSE?

Let $ a_n = \dfrac{(-1)^{n}}{\sqrt{1+n}} $ and let $ c_n = \sum_{k=0}^{n} a_{n-k} a_k $, where $ n \in \mathbb{N} \cup \{0\} $. Then which one of the following is TRUE?

Let $ a_n = n + \frac{1}{n} $, $ n \in \mathbb{N} $. Then the sum of the series $ \sum_{n=1}^{\infty} (-1)^{n+1} \dfrac{a_{n+1}}{n!} $ is

Let $ a, b, c \in \mathbb{R} $. Which of the following values of $ a, b, c $ do NOT result in the convergence of the series
\[ \sum_{n=3}^{\infty} \frac{a^n}{n^b (\log_e n)^c} ? \]

$$\text{Let } a_n = \begin{cases} 2 + \dfrac{(-1)^{\frac{n-1}{2}}}{n}, & \text{if } n \text{ is odd} \\ 1 + \dfrac{1}{2^n}, & \text{if } n \text{ is even} \end{cases}, \quad n \in \mathbb{N}.$$
Then which one of the following is TRUE?

Let $ s_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} $ for $ n \in \mathbb{N} $. Then which one of the following is TRUE for the sequence $ \{ s_n \}_{n=1}^{\infty} $?

Let $ a_n = \dfrac{b_{n+1}}{b_n} $, where $ b_1 = 1 $, $ b_2 = 1 $, and $ b_{n+2} = b_n + b_{n+1} $, $ n \in \mathbb{N} $. Then $ \lim_{n \to \infty} a_n $ is

Let $ a_n = \dfrac{(1 + (-1)^{n})}{2^n} + \dfrac{(1 + (-1)^{n-1})}{3^n}. $ Then the radius of convergence of the power series $ \sum_{n=1}^{\infty} a_n x^n \text{ about } x = 0 \text{ is ..............} $

The coefficient of $ x^4 $ in the power series expansion of $ e^{\sin x} $ about $ x = 0 $ is ............. (correct up to three decimal places).

Let $ a_k = (-1)^{k-1} $, $ s_n = a_1 + a_2 + \cdots + a_n $, and $ \sigma_n = \frac{1}{n} (s_1 + s_2 + \cdots + s_n) $, where $ k, n \in \mathbb{N} $.

\[ \text{ Then} \,\, \lim_{n \to \infty} \sigma_n = \text{.................} \text{ (correct up to one decimal place)}. \]

Let $ f : \mathbb{R} \to \mathbb{R} $ be such that $ f'' $ is continuous on $ \mathbb{R} $ and $ f(0) = 1 $, $ f'(0) = 0 $, and $ f''(0) = -1 $. \[ \text{Then} \,\, \lim_{x \to \infty} \left( f \left( \sqrt{\frac{2}{x}} \right) \right)^x \text{ is ............} \text{ (correct up to three decimal places)}. \]

Let $ f(x) = \sum_{n=0}^{\infty} (-1)^n x(x - 1)^n $ for $ 0 < x < 2 $. Then the value of $ f \left( \frac{\pi}{4} \right) $ is .............

For $ x > -\frac{1}{2} $, let $ f_1(x) = \frac{2x{1+2x} $, $ f_2(x) = \log_e(1 + 2x) $ and $ f_3(x) = 2x $. Then which one of the following is TRUE?}

Let $ f : \mathbb{R} \setminus \{0\} \to \mathbb{R} $ be defined by $ f(x) = x + \frac{1{x^3} $. On which of the following interval(s) is $ f $ one-to-one?}

Let $ f : \mathbb{R} \to \mathbb{R} $ be a function and let $ I $ be a bounded open interval in $ \mathbb{R} $. Define
\[ W(f, I) = \sup \{ f(x) | x \in I \} - \inf \{ f(x) | x \in I \} \] Which one of the following is FALSE?