List of top Real Analysis Questions

Let \( f(x) = 2x^3 - 9x^2 + 7. \) Which of the following is true?

Let \( s_n = 1 + \dfrac{(-1)^n}{n}, \, n \in \mathbb{N}. \) Then the sequence \(\{s_n\}\) is
 

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( f, f', f'' \) are continuous with \( f > 0, f' > 0, f'' > 0. \) Then \( \displaystyle \lim_{x \to -\infty} \frac{f(x) + f'(x)}{2} \) is ............
 

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 ..............
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 .............
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 \( k, n \in \mathbb{N} \). 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 \( f''(0) = -1 \). Then \[ \lim_{x \to \infty} \left( f \left( \sqrt{\frac{2}{x}} \right) \right)^x \text{ is ............} \text{ (correct up to three decimal places)}. \]
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 \[ a_n = \frac{(1 + (-1)^{n})}{2n} + \frac{(1 + (-1)^{n-1})}{3n}. \] Then the radius of convergence of the power series \[ \sum_{n=1}^{\infty} a_n x^n \text{ about } x = 0 \text{ is ..............} \]
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?
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
\[ \sum_{n=3}^{\infty} \frac{a^n}{n^b (\log n)^c} ? \]
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 = \left\{ \begin{array}{ll} 2 + (-1)^{n \frac{n}{2^n} & \text{if } n \text{ is odd}
1 + \frac{n}{2^n} & \text{if } n \text{ is even}, \end{array} \right. \ n \in \mathbb{N}.}
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 \( \sum_{n=1}^{\infty} (-1)^n n^{n+1 \frac{a_{n+1}}{n!} \) is}
Let \( a_n = \frac{(-1)^{n}{\sqrt{1+n}} \) 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 \( s_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} \) for \( n \in \mathbb{N} \). Then which one of the following is TRUE for the sequence \( \{ s_n \} \)?
Let \( a_n = \frac{b_{n+1}}{b_n} \), 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