List of top Real Analysis Questions on Sequences and Series of real numbers asked in IIT JAM MA

Let $ f(x) = \sqrt{x} + \alpha x, \; x > 0 $ and \[ g(x) = a_0 + a_1(x - 1) + a_2(x - 1)^2 \] be the sum of the first three terms of the Taylor series of $ f(x) $ around $ x = 1 $. If $ g(3) = 3 $, then $ \alpha $ is .............

Consider the expansion of the function $ f(x) = \dfrac{3}{(1 - x)(1 + 2x)} $ in powers of $ x $, valid in $ |x| < \dfrac{1}{2}. $ Then the coefficient of $ x^4 $ is ................

Suppose that $ S $ is the sum of a convergent series $ \displaystyle \sum_{n=1}^{\infty} a_n. $ Define $ t_n = a_n + a_{n+1} + a_{n+2}. $ Then the series $ \displaystyle \sum_{n=1}^{\infty} t_n $

Define $ s_1 = \alpha > 0 $ and $ s_{n+1} = \sqrt{\dfrac{1 + s_n^2}{1 + \alpha}}, \, n \geq 1. $ Which of the following is true?

Let $\{a_n\}$ be a sequence of positive real numbers. Suppose that $l = \lim_{n \to \infty} \dfrac{a_{n+1}}{a_n}.$ Which of the following is true?

The radius of convergence of the power series $ \displaystyle \sum_{n=1}^{\infty} \left( \dfrac{n+2}{n} \right)^{n^2} x^n $ is

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

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