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

\[ \lim_{n\to\infty}\left(\frac{n^2+1}{\sqrt{n^6+1}}+\cdots+\frac{n^2+n}{\sqrt{n^6+n}}\right) = \underline{} \] rounded off to one decimal place.

Which of the following statements is/are TRUE?

For each positive integer \( n \), let \[ x_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \log n \]
and \[ y_n = \int_1^n \frac{\cos t}{t^2} \, dt. \]
Which ONE of the following statements about the sequences \( (x_n) \) and \( (y_n) \) is TRUE?

The sum of the infinite series \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \pi^{2n+1}}{2^{2n+1} (2n)!} \] is equal to

For \( n \in \mathbb{N} \), define \( x_n \) and \( y_n \) by \[ x_n = (-1)^n \cos \frac{1}{n} \quad \text{and} \quad y_n = \sum_{k=1}^{n} \frac{1}{n + k}. \] Then, which one of the following is TRUE?

The value of the infinite series \[ \sum_{n=1}^{\infty} n \left( \frac{3}{4} \right)^{2n-1} \] is equal to ............. (rounded off to two decimal places).

The radius of convergence of the power series \[ \sum_{n=1}^{\infty} \frac{(x + \frac{1}{4})^n}{(-2)^n n^2} \] about \( x = -\frac{1}{4} \) is equal to ............ (rounded off to two decimal places).

Let \( \alpha \) be the real number such that \[ \lim_{x \to 0} \frac{(1 - \cos x)(22x^2 + x - 4)}{x^3} = \alpha \ln 2. \] Then, the value of \( \alpha \) is equal to ............ (rounded off to two decimal places).

For \( n \in \mathbb{N} \), let \[ x_n = \sum_{k=1}^{n} \frac{k}{n^2 + k}. \] Then, which of the following is/are TRUE?

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

Let \( x_1 = 1 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = \left( \frac{1}{2} + \frac{\sin^2 n}{n} \right) x_n. \] Then, which one of the following is TRUE?

Let \( x_1>0 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = x_n + 4. \] If \[ \lim_{n \to \infty} \left( \frac{1}{x_1 x_2 x_3} + \frac{1}{x_2 x_3 x_4} + \cdots + \frac{1}{x_{n+1} x_{n+2} x_{n+3}} \right) = \frac{1}{24}, \] then the value of \( x_1 \) is equal to:

Define the sequence \[ s_n = \begin{cases} \dfrac{1}{2^n}\displaystyle\sum_{j=0}^{n-2} 2^{2j}, & \text{if } n \text{ is even and } n \gt 0, \\[8pt] \dfrac{1}{2^n}\displaystyle\sum_{j=0}^{n-1} 2^{2j}, & \text{if } n \text{ is odd and } n \gt 0. \end{cases} \] Define \[ \sigma_m = \frac{1}{m}\sum_{n=1}^{m} s_n. \] The number of limit points of the sequence \(\{\sigma_m\}\) is _________.

Consider the two series \[ \text{I. } \sum_{n=1}^{\infty} \frac{1}{n^{1 + (1/n)}} \quad \text{and} \quad \text{II. } \sum_{n=1}^{\infty} \frac{1}{n(\ln n)^{1/n}}. \] Which one of the following holds?

Let \( f : \mathbb{N} \to \mathbb{N} \) be a bijective map such that \[ \sum_{n=1}^{\infty} \frac{f(n)}{n^2}<+\infty. \] The number of such bijective maps is

Let \( a_n \) be the sequence given by \[ a_n = \max \left( \sin \left( \frac{n \pi}{3} \right), \cos \left( \frac{n \pi}{3} \right) \right), \quad n \geq 1. \] Then which of the following statements is/are TRUE about the subsequences \( \{a_{6n-1}\} \) and \( \{a_{6n+1}\} \)?

Let \( a_1 = b_1 = 0 \), and for each \( n \geq 2 \), let \( a_n \) and \( b_n \) be real numbers given by \[ a_n = \sum_{m=2}^{n} (-1)^{m} m (\log(m))^m \] \[ b_n = \sum_{m=2}^{n} \frac{1}{(\log(m))^m}. \] Then which one of the following is TRUE about the sequences \( \{a_n\} \) and \( \{b_n\} \)?