Question:
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
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
Updated On: Oct 1, 2024
- \( (a_n) \) is a Cauchy sequence but \( (b_n) \) is NOT a Cauchy sequence
- \( (a_n) \) is NOT a Cauchy sequence but \( (b_n) \) is a Cauchy sequence
- both \( (a_n) \) and \( (b_n) \) are Cauchy sequences
- neither \( (a_n) \) nor \( (b_n) \) is a Cauchy sequence
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
The correct option is (B): \( (a_n) \) is NOT a Cauchy sequence but \( (b_n) \) is a Cauchy sequence
Was this answer helpful?
0
0
Top Questions on Sequences and Series of real numbers
- Which of the following functions represents a cumulative distribution function?
- IIT JAM MS - 2024
- Mathematics
- Sequences and Series of real numbers
- 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?
- IIT JAM MS - 2024
- Mathematics
- Sequences and Series of real numbers
- Let {ππ }πβ₯1 be a sequence such that π1=1 and 4ππ+1=\(\sqrt{45 + 16π_π} ,π\)=1, 2, 3, β¦ . Then, which one of the following statements is TRUE?
- IIT JAM MS - 2023
- Mathematics
- Sequences and Series of real numbers
- Let the series π and π be defined by
\(β^β_{n-0}\frac{ 2 β 5 β 8 β― (3π + 2) }{1 β 5 β 9 β― (4π + 1)}\) and \(β^β _{n=1}(1 + \frac{1 }{π} ) ^{βπ^2}\)
respectively. Then, which one of the following statements is TRUE?- IIT JAM MS - 2023
- Mathematics
- Sequences and Series of real numbers
- Let π1, π2,be a sequence of π. π. π. random variables, each having the probability density function
\(f(x) =\begin{cases} \frac{x^2r^{-x}}{2} & \quad \text{if }x β₯0,\\ 0, & \quad Otherwise \end{cases}\)
For some real constants π½, πΎ and π, suppose that
\(lim_{ββ} ( \frac{1}{ π} β^n_{i=1}π_πβ€ π₯) \)=\(\begin{cases} 0 & \quad if\,x< Ξ².\\ kx, & \quad if\,Ξ²β€xβ€y.\\ ky, & \quad if\,x>y.\end{cases}\)
Then, the value of 2π½ + 3πΎ + 6π equals _______- IIT JAM MS - 2023
- Mathematics
- Sequences and Series of real numbers
View More Questions
Questions Asked in IIT JAM MS exam
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- IIT JAM MS - 2024
- Probability
- Let the random vector \( (X, Y) \) have the joint probability density function
\[f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.\]
Then \( \text{Cov}(X, Y) \) is equal to:- IIT JAM MS - 2024
- Probability
- Two factories \( F_1 \) and \( F_2 \) produce cricket bats that are labelled. Any randomly chosen bat produced by factory \( F_1 \) is defective with probability 0.5 and any randomly chosen bat produced by factory \( F_2 \) is defective with probability 0.1. One of the factories is chosen at random, and two bats are randomly purchased from the chosen factory. Let the labels on these purchased bats be \( B_1 \) and \( B_2 \). If \( B_1 \) is found to be defective, then the conditional probability that \( B_2 \) is also defective is equal to __________ (round off to 2 decimal places).
- IIT JAM MS - 2024
- Probability
- Consider a coin for which the probability of obtaining head in a single toss is \( \frac{1}{3} \). Sunita tosses the coin once. If head appears, she receives a random amount of \( X \) rupees, where \( X \) has the \( \text{Exp}\left( \frac{1}{9} \right) \) distribution. If tail appears, she loses a random amount of \( Y \) rupees, where \( Y \) has the \( \text{Exp}\left( \frac{1}{3} \right) \) distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Probability
- If 12 fair dice are independently rolled, then the probability of obtaining at least two sixes is equal to __________ (round off to 2 decimal places)
- IIT JAM MS - 2024
- Probability
View More Questions