List of top Mathematics Questions on Sequences and Series of real numbers

Let \(\{a_n\}_{n\geq 1}\) be a sequence of real numbers given by

Let \(f:\mathbb{N}\to\{1,2,3,\ldots,100\}\) be an onto function. Consider the following statements.

\[ \lim_{n\to\infty}\left(\frac{n^2+1}{\sqrt{n^6+1}}+\cdots+\frac{n^2+n}{\sqrt{n^6+n}}\right) = \underline{} \] 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:

Let \( a_1 = 1 \), \( a_{n+1} = a_n \left( \frac{\sqrt{n} + \sin n}{n} \right) \), and \( b_n = a_n^2 \) for all \( n \in \mathbb{N} \). Then which of the following statements is/are correct?

Let \( a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) and \( b_n = \sum_{k=1}^{n} \frac{1}{k^2} \) for all \( n \in \mathbb{N} \). Then

Which of the following functions represents a cumulative distribution function?

Let {𝑎_𝑛 }_𝑛≥1 be a sequence such that 𝑎₁=1 and 4𝑎_𝑛+1=\(\sqrt{45 + 16𝑎_𝑛} ,𝑛\)=1, 2, 3, … . Then, which one of the following statements is TRUE?

Let the series 𝑆 and 𝑇 be defined by
\(∑^∞_{n-0}\frac{ 2 ⋅ 5 ⋅ 8 ⋯ (3𝑛 + 2) }{1 ⋅ 5 ⋅ 9 ⋯ (4𝑛 + 1)}\) and \(∑^∞ _{n=1}(1 + \frac{1 }{𝑛} ) ^{−𝑛^2}\)
respectively. Then, which one of the following statements is TRUE?

Let 𝑋1, 𝑋2,be a sequence of 𝑖. 𝑖. 𝑑. random variables, each having the probability density function
\(f(x) =\begin{cases}   \frac{x^2r^{-x}}{2}       & \quad \text{if }x ≥0,\\  0, & \quad Otherwise \end{cases}\)
For some real constants 𝛽, 𝛾 and 𝑘, suppose that 
\(lim_{→∞} ( \frac{1}{ 𝑛} ∑^n_{i=1}𝑋_𝑖≤ 𝑥) \)=\(\begin{cases}   0     & \quad if\,x< β.\\  kx, & \quad if\,β≤x≤y.\\  ky, & \quad if\,x>y.\end{cases}\)
Then, the value of 2𝛽 + 3𝛾 + 6𝑘 equals _______

Let
\(𝑆_𝑛 =∑_{k=1}^n\frac{1+k2^k}{4^{k-1}},n=1,2,...\)
Then, lim𝑛_→∞ 𝑆_𝑛 equals (round off to two decimal places)

Let {an}_n≥1 be a sequence of non-zero real numbers. Then which one of the following statements is true ?

Let {a_n}_n≥1 be a sequence of real numbers such that \(a_n=\frac{1}{3^n}\) for all n ≥ 1. Then which of the following statements is/are true ?

Let {a_n}_n≥1 be a sequence of real numbers such that a_1+5m = 2, a_2+5m = 3, a_3+5m = 4, a_4+5m = 5, a_5+5m = 6, m = 0, 1, 2, … . Then limsup_n→∞ a_n + liminf_n→∞ a_n equals __________

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 _________.