List of top Real Analysis Questions

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)

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 p(x) = x⁵⁷ + 3x¹⁰ - 21x³ + x² + 21 and
\(q(x)=p(x)+\sum\limits_{j=1}^{57}p^{(j)}(x) \ \text{for all }x \in \R,\)
where p^(j)(x) denotes the j^th derivative of p(x). Then the function q admits

Let f(x) = cos(x) and g(x) =\(1-\frac{x^2}{2}\) for  \(x \in (-\frac{\pi}{2},\frac{\pi}{2})\). Then

Let f : \(\R → \R\) be an infinitely differentiable function such that f" has exactly two distinct zeroes. Then

Let f : (−1, 1) → \(\R\) be a differentiable function satisfying f(0) = 0. Suppose there exists an M > 0 such that |f' (x)| ≤ M|x| for all x ∈ (−1, 1). Then

Let f(x) = \(\sqrt[3]{x}\) for x ∈ (0, ∞), and θ(h) be a function such that
f(3 + h) − f(3) = hf′ (3 + θ(h)h)
for all h ∈ (−1, 1). Then \(\lim\limits_{h→0} θ(h)\) is equal to _________. (rounded off to two decimal places)

The global minimum value of
f(x) = |x - 1| + |x - 2|²
on \(\R\) 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?

The value of
\[ \lim_{n \to \infty} \sum_{k=2}^n \frac{\sqrt{n} + 1 - \sqrt{n}}{k(\ln k)^2} \]
is equal to

The sum of the series
\(∑^∞_{ n=1} \frac{1}{(4n-3)(4n+1) }\) 
is equal to_____.(Round off to two decimal places)

Let r be the radius of convergence of the power series
\(\frac{1}{3}+\frac{x}{5}+\frac{x^2}{3^2}+\frac{x^3}{5^2}+\frac{x^4}{3^3}+\frac{x^5}{5^3}+\frac{x^6}{3^4}+\frac{x^7}{5^4}+...\)
Then the value of r² is equal to _________. (Rounded off to two decimal places)

Consider the system of linear equations\[ \begin{cases}x + y + t = 4, \\2x - 4t = 7, \\x + y + z = 5, \\x - 3y - z - 10t = \lambda,\end{cases}\]where \( x, y, z, t \) are variables and \( \lambda \) is a constant. Then which one of the following is true?

For t ∈ \(\R\), let [𝑡] denote the greatest integer less than or equal to t.
Let D = {(x, y) ∈ \(\R^2\) ∶ x2 + y2 < 4}. Let f : D → \(\R\) and g : D → \(\R\) be defined by f(0, 0) = g(0, 0) = 0 and
\(f(x,y)=[x^2+y^2]\frac{x^2y^2}{x^4+y^4},\ \ \ g(x,y)=[y^2]\frac{xy}{x^2+y^2}\)
for (x, y) ≠ (0, 0). Let E be the set of points of D at which both f and g are discontinuous. The number of elements in the set E is _________.

Let (x_n) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?

Define f: [0,1] → [0,1] by
\(f(x) = \begin{cases} 1 & \text{if } x =0 \\ \frac{1}{n} & \text{if } x=\frac{m}{n}\ \text{for some}\ m, n \isin \N\ \text{with}\ m\le n\ \text{and gcd(m, n)} = 1, \\ 0 & \text{if} x\isin[0,1]\ \text{is irrational}\end{cases}\)
and define g: [0,1]→ [0,1] by
\(g(n) = \begin{cases} 0 & \text{if } x=0 \\ 1 & \text{if } x\isin(0,1]. \end{cases}\)
Then which of the following is/are true?

A real-valued function y(x) defined on \(\R\) is said to be periodic if there exists a real number \(T\gt0\) such that y(x + T) = y(x) for all \(x\isin\R\). Consider the differential equation
\(\frac{d^2y}{dx^2}+4y=\sin(ax), x\isin\R, \quad (*)\)
where \(a\isin\R\) is a constant. 
Then which of the following is/are true?

Let V be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 5, together with the zero polynomial. Let T: V→ \(\R\) be the linear map defined by 7(1) = 1 and T(x(x-1)... (x-k+1)) = 1 for 1 ≤ k ≤ 5.
Then which of the following is/are true?