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

For n ≥ 3, let a regular n-sided polygon P_n be circumscribed by a circle of radius R_n and let r_n be the radius of the circle inscribed in P_n. Then
\(\lim\limits_{n \rightarrow \infin}(\frac{R_n}{r_n})^{n^2}\)
equals

For \(n\ \in \N\), let
\(a_n=\frac{1}{(3n+2)(3n+4)}\) and \(b_n=\frac{n^3+\cos(3^n)}{3n+n^3}\).
Then, which one of the following is TRUE ?

Define the sequences \(\left\{a_n\right\}^{\infin}_{n=3}\) and \(\left\{b_n\right\}^{\infin}_{n=3}\) as
\(a_n=(\log n+\log\ \log n)^{\log n}\) and \(b_n=n^{(1+\frac{1}{\log n})}\).
Which one of the following is TRUE ?

For p, q, r ∈ ℝ, r ≠ 0 and n ∈ \(\N\), let
\(a_n=p^n n^q(\frac{n}{n+2})^{n^2}\) and \(b_n=\frac{n^n}{n!r^n}(\sqrt{\frac{n+2}{n}})\).
Then, which one of the following statements is TRUE ?

Let \(\left\{a_n\right\}^{\infin}_{n=1}\) be a sequence of real numbers.
Then, which of the following statements is/are always TRUE ?

Which of the following statements is/are TRUE ?

Define f: ℝ → ℝ and g: ℝ → ℝ as follows
\(f(x)=\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m!)^2}\) and \(g(x)=\frac{x}{2}\sum\limits^{\infin}_{m=0}\frac{(-1)^mx^{2m}}{2^{2m}(m+1)!m!}\) for x ∈ \(\R\).
Let x₁, x₂, x₃, x₄ ∈ ℝ be such that 0 < x₁ < x₂ , 0 < x₃ < x₄,
f(x₁) = f(x₂) = 0,    f(x) ≠ 0 when x₁ < x < x₂,
g(x₃) = g(x₄) = 0 and g(x) ≠ 0 when x₃ < x < x₄.
Then, which of the following statements is/are TRUE ?

For 0 < a < 4, define the sequence \(\left\{x_n\right\}^{\infin}_{n=1}\) of real numbers as follows :
x₁ = a and x_n+1 + 2 = −x_n(x_n - 4) for n ∈ \(\N\).
Which of the following is/are TRUE ?

Let a₁ = 1, b₁ = 2 and c₁ = 3. Consider the convergent sequences
\(\left\{a_n\right\}^{\infin}_{n=1},\left\{b_n\right\}^{\infin}_{n=1} \text{ and }\left\{c_n\right\}^{\infin}_{n=1}\)
defined as follows :
\(a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\frac{b_n+c_n}{2} \text{ and } c_{n+1}=\frac{c_n+a_n}{2} \text{ for } n \ge1.\)
Then,
\(\sum\limits_{n=1}^{\infin}b_nc_n(a_{n+1}-a_n)+\sum\limits_{n=1}^{\infin}(b_{n+1}c_{n+1}-b_nc_n)a_{n+1}\)
equals _____________ (rounded off to two decimal places)

For n ∈ \(\N\), let
\(a_n=\frac{1}{n^{n-1}}\sum\limits_{k=0}^n\frac{n!}{k!(n-k)!}\frac{n^k}{k+1}\)
and \(\beta=\lim\limits_{n\rightarrow \infin}a_n\). Then, the value of log 𝛽 equals ___________ (rounded off to two decimal places).

Define the function \(f:(-1,1)\rightarrow (-\frac{\pi}{2},\frac{\pi}{2})\) by
\(f(x)=\sin^{-1}x\)
Let a₆ denote the coefficient of x⁶ in the Taylor series of (f(x))² about x = 0. Then, the value of 9a₆ equals ____________ (rounded off to two decimal places).

Let (a_n) be a sequence of real numbers such that the series \(\sum\limits_{n=0}^{\infin}a_n(x-2)^n\) converges at x = −5. Then this series also converges at

Let \(a_n=\left(1+\frac{1}{n}\right)^n\) and \(b_n=n\cos(\frac{n!\pi}{2^{10}})\) for n ∈ \(\N\). Then

Let (an) be a sequence of real numbers defined by
\(a_n=\begin{cases} 1 & \text{if } n \text{ is prime}\\    -1 & \text{if } n \text{ is not prime} \end{cases}\)
Let \(b_n=\frac{a_n}{n}\) for n ∈ \(\N\). Then

Let \(a_n=\sin(\frac{1}{n^3})\) and \(b_n=\sin(\frac{1}{n})\) for n ∈ \(\N\). Then

Let \(a_n=\frac{1+2^{-2}+...+n^{-2}}{n}\) for n ∈ \(\N\). Then

Let (a_n) and (b_n) be sequences of real numbers such that
\(|a_n-a_{n+1}|=\frac{1}{2^n}\) and \(|b_n-b_{n+1}|=\frac{1}{\sqrt{n}}\) for n ∈ \(\N\).
Then

Let S be the set of all real numbers α such that the solution y of the initial value problem
\(\frac{dy}{dx}=y(2-y),\\y(0)=\alpha,\)
exists on [0, ∞). Then the minimum of the set S is equal to __________. (rounded off to two decimal places)

Let R₁ and R₂ be the radii of convergence of the power series \(\sum\limits_{n=1}^{\infin}(-1)^nx^{n-1}\) and \(\sum\limits_{n=1}^{\infin}(-1)^n\frac{x^{n+1}}{n(n+1)}\), respectively. Then

The value of \(\lim\limits_{n \rightarrow \infin}\left(1+\frac{1}{2^n}+\frac{1}{3^n}+...+\frac{1}{(2023)^n}\right)^\frac{1}{n}\) is equal to _____________ . (rounded off to two decimal places)

The radius of convergence of the power series \[\sum_{n=1}^{\infty} \left(\frac{n^3}{4^n}\right) x^{5n}\] is

Let \( (x_n) \) and \( (y_n) \) be sequences of real numbers defined by
\[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. \]
Then which one of the following is true?

Suppose \[ a_n = \frac{3^n + 3}{5^n - 5} \quad \text{and} \quad b_n = \frac{1}{(1 + n^2)^{1/4}} \quad \text{for} \ n = 2, 3, 4, \ldots \] Then which one of the following is true?

Consider the series \[\sum_{n=1}^{\infty} n^m \left(\frac{1}{1 + \frac{1}{n^p}}\right)\] where \( m \) and \( p \) are real numbers. Under which of the following conditions does the above series converge?

Let \( T \) denote the sum of the convergent series
\[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots + (-1)^{n+1} \frac{1}{n} + \ldots\]
and let \( S \) denote the sum of the convergent series
\[ 1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \frac{1}{12} + \sum_{n=1}^{\infty} a_n\]
where
\[ a_{3m-2} = \frac{1}{2m-1} , a_{3m-1} = 0, \text{ and } a_{3m} = \frac{-1}{4m} \text{ for } m \in \mathbb{N}.\]
Then which one of the following is true?