English
Hindi

Question

Download Solution PDF

Given (an)n≥1 a sequence of real numbers, which of the following statements is true?

This question was previously asked in
CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
Download PDF Attempt in App
View all CSIR NET Papers >
  1.  converges 
  2. There is a subsequence  such that converges
  3. There is a number b such that  converges
  4. There is a number b and a subsequence such that  converges

Answer (Detailed Solution Below)

Option 4 : There is a number b and a subsequence such that  converges
Free Tests
View all Free tests >
Free
Seating Arrangement
10 Qs. 20 Marks 15 Mins
Start Now

Detailed Solution

Download Solution PDF

Explanation:

Tips: Try to discard options by taking suitable choice for < an >.

option (1). Let an = 1 then (-1)n . ≠ 0

option (2). Let an = <1> and  = <1> then

  Not convergent.

option (3), option (4): (NB: may be you have to try with more) then one sea, <an>

Let an = (-1)n then

But here fixed 'b S.t above series become cgt. You may take b = ½ or = -½ but not both otherwise uniqueness will be lost.

⇒ option (3) is false. 

option (4): As discussed earlier. take b = ½ and  = <1> then about series becomes convergent. Hence option (4) is true.

Download Solution PDF
Latest CSIR NET Updates

Last updated on Jun 5, 2025

-> The NTA has released the CSIR NET 2025 Notification for the June session.

-> The CSIR NET Application Form 2025 can be submitted online by 23rd June 2025

-> The CSIR UGC NET is conducted in five subjects -Chemical Sciences, Earth Sciences, Life Sciences, Mathematical Sciences, and Physical Sciences. 

-> Postgraduates in the relevant streams can apply for this exam.

-> Candidates must download and practice questions from the CSIR NET Previous year papers. Attempting the CSIR NET mock tests are also very helpful in preparation.

‹‹ Previous Ques
Next Ques ››
Exams Tests Super Practice

More Sequences & Series (Convergence) Questions

Q1.Let (an)n≥1 be a bounded sequence in ℝ. Which of the following statements is FALSE? 
Q2.Consider the series \(\rm\displaystyle\sum_{n=1}^{\infty}\) an, where an = (−1)n+1\(\rm (\sqrt{n+1}−\sqrt{n})\). Which of the following statements is true?

More Analysis Questions

Q1.What is the cardinality of the set of real solutions of ex + x = 1?
Q2.Let u = u(x, t) be the solution of the following initial value problem  \(\rm \left\{\begin{matrix}u_t+2024u_x=0,&x ∈ R, t>0\\\ u(x, 0)=u_0(x), &x ∈ R\end{matrix}\right.\) where u0 : ℝ → ℝ s an arbitrary C1 function. Consider the following statements:  S1 : If At = {x ∈ ℝ : u(x, t) < 1} and |At| denotes the Lebesgue measure of A, for every t ≥ 0, then |At| = |A0|, ∀t > 0 S2 : If u0 is Lebesgue integrable, then for every t > 0, the function x → u(x, t) is Lebesgue integrable. 
Q3.Let (an)n≥1 be a bounded sequence in ℝ. Which of the following statements is FALSE? 
Q4.Consider the set A = {x ∈ ℚ : 0 < (√2 - 1)x < √2 + 1} as a subset of ℝ. Which of the following statements is true?
Q5.For each n ≥ 1 define fn : ℝ → ℝ by \(\rm f_n(x)=\frac{x^2}{√{x^2+\frac{1}{n}}}, \) x ∈ ℝ where √ denotes the non-negative square root. Wherever \(\rm \lim_{n \rightarrow \infty}f_n(x)\) exists, denote it by f(x). Which of the following statements is true? 
Q6.Let \(\rm S =\left\{x \in R:x >1 \ and \ \frac{1-x^4}{1-x^3}>22\right\}\) Which of the following is true about S? 
Q7.Let S be a dense subset of R and f : ℝ → ℝ a given function. Define g : S → ℝ by g(x) = f(x). Which of the following statements is necessarily true? 
Q8.Let C be the collection of all sets S such that the power set of S is countably infinite. Which of the following statements is true? 
Q9.Let 𝑐00 = {(𝑥1, 𝑥2, 𝑥3, … ) ∶ 𝑥𝑖 ∈ ℝ, 𝑖 ∈ ℕ, 𝑥𝑖 ≠ 0 only for finitely many indices 𝑖}. For (𝑥1, 𝑥2, 𝑥3, … ) ∈ 𝑐00, let ‖(𝑥1, 𝑥2, 𝑥3, … )‖∞ = sup{|𝑥𝑖 | ∶ 𝑖 ∈ ℕ}. Define 𝐹, 𝐺 ∶ (𝑐00, ‖⋅‖∞) → (𝑐00, ‖⋅‖∞) by 𝐹((𝑥1, 𝑥2, … , 𝑥𝑛, … )) = \(\rm \left((1+x)_{x_1}, (2+\frac{1}{2})x_2,..., (n+\frac{1}{n})x_n,...\right)\) 𝐺((𝑥1, 𝑥2, … , 𝑥𝑛, … )) = \(\rm \left(\frac{x_1}{1+1},\frac{x_2}{2+\frac{1}{2}},..., \frac{x_n}{n+\frac{1}{n}},..\right)\) for all (𝑥1, 𝑥2, … , 𝑥𝑛, … ) ∈ 𝑐00. Then
Q10.Let ℓ2 = {(𝑥1, 𝑥2, 𝑥3, … ) ∶ 𝑥𝑛 ∈ ℝ for all 𝑛 ∈ ℕ and \(\rm \Sigma_{n=1}^\infty x_n^2<\infty \}\) For a sequence (𝑥1, 𝑥2, 𝑥3, … ) ∈ ℓ2 , define ‖(𝑥1, 𝑥2, 𝑥3, … )‖2 = \(\rm (\Sigma_{n=1}^\infty x_n^2)^{\frac{1}{2}}\) Let 𝑆 ∶ (ℓ2 , ‖⋅‖2) → (ℓ2 , ‖⋅‖2) and 𝑇 ∶ (ℓ2 , ‖⋅‖2) → (ℓ2 , ‖⋅‖2) be defined by 𝑆(𝑥1, 𝑥2, 𝑥3, … ) = (𝑦1, 𝑦2, 𝑦3, … ), where 𝑦𝑛 = \(\rm \left\{\begin{matrix}0,&n=1\\\ x_{n-1},& n\ge2\end{matrix}\right.\) 𝑇(𝑥1, 𝑥2, 𝑥3, … ) = (𝑦1, 𝑦2, 𝑦3, … ), where 𝑦𝑛 = \(\rm \left\{\begin{matrix}0,&n\ is\ odd\\\ x_{n},& n\ is\ even\end{matrix}\right.\) Then

More Mathematical Science Questions

Q1.The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 
Q2.What is the cardinality of the set of real solutions of ex + x = 1?
Q3.Let A be a 10 x 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true? 
Q4.Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 
Q5.Consider the contour γ given by \(\rm \gamma (\theta)=\left\{\begin{matrix}e^{2i \theta}&for\ \theta \in [0, \pi /2]\\\ 1+2e^{2i\theta}&for\ \theta \in [\pi/2, 3\pi/2]\\\ e^{2i\theta}&for\ \theta \in [3\pi/2, 2\pi]\end{matrix}\right.\) Then what is the value of \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q6.Let X1....X10 be a random sample from a distribution with the probability density function \(\rm f(x|θ)=\left\{\begin{matrix}θ x^{θ-1}, & if \:0<x<1\\\ 0, & otherwise,\end{matrix}\right.\) where θ > 0 Is an unknown parameter. The prior distribution of θ is given by   \(\rm \pi (θ)=\left\{\begin{matrix}θ e^{-θ}, & if\ θ>0\\\ 0, & otherwise,\end{matrix}\right.\) The Bayes estimator of θ under squared error loss is  
Q7.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
Q8.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q9.Let X1, X2 be a random sample from a population having probability density function f ∈ { f0, f1} where  \(\rm f_0(x)=\left\{\begin{matrix}\frac{1}{2}& if\ 0\le x\le 2\\\ 0&otherwise\end{matrix}\right.\ and \ \rm f_1(x)=\left\{\begin{matrix}\frac{1}{4}& if\ 0\le x\le 4\\\ 0&otherwise\end{matrix}\right.\) For testing the null hypothesis H0 : f = f0 against the alternate hypothesis H1 : f = f1, the power of a most powerful test of size α = 0.05 is equal to  
Q10.Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where \(\rm A=\begin{pmatrix}2&0\\\ 0&1\end{pmatrix}\). What is the characteristic polynomial of T? 
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
support@testbook.com
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Hot Links： teen patti - 3patti cards game teen patti all games real teen patti