Select Page

# What are we learning?

Sequences & Subsequences are the key features in the filed of real analysis. We will see how to imply these concepts in our problem

# Understand the problem

Let $s_n$ = 1+$\frac{1}{1!}$+$\frac{1}{2!}$+……..+$\frac{1}{n!}$ for n $\in$ $\mathbb{N}$ Then which of the following is TRUE for the sequence {$s_{n}\}^\infty_{n=1}$:   (a) {$s_{n}\}^\infty_{n=1}$ converges in $\mathbb{Q}$.   (b) {$s_{n}\}^\infty_{n=1}$ is a Cauchy sequence but does not converges to $\mathbb{Q}$.   (c) The subsequence  {$s_{k^n}\}^\infty_{n=1}$ is convergent in $\mathbb{R}$ when k is a even natural number.   (d) {$s_{n}\}^\infty_{n=1}$ is not a Cauchy sequence.
IIT Jam 2018
Easy
##### Suggested Book
 Calculus: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability – Vol 2 Tom M. Apostol

Do you really need a hint? Try it first!

I am going to give you 3 clues in the beginning you try to work out using them. Then I will elaborate this clues in the following hints  (I) Every convergent sequence is a Cauchy sequence  (II)Every subsequence of a convergent sequence is convergent  (III)Consider then term 1+$\frac{1}{1!}$+$\frac{1}{2!}$+……..+$\frac{1}{n!}$ Does this remind you any well known series?

I wil start with (III) consider $e^x$=1+$\frac{x}{1!}$+$\frac{x^2}{2!}$+……..+$\frac{x^n}{n!}$ Isn’t the seris that we have to , is the value at x=1. Hence the given series$\rightarrow$ e $\in$ $\mathbb{R}$ \ $\mathbb{Q}$

So option (a) is incorrect.

Every subsequence of a convergent sequence is convergent so {$s_{k^n}\}^\infty_{n=1}$ is convergent not only for even k, but for any $k \in \Bbb N$. So option (c) is incorrect.

Every convergent sequence is a Cauchy sequence so option (d) is incorrect and $e \in$ $\mathbb{R}$ so the given subsequence is convergent in $\mathbb{R}$. So only option (b) is correct.

# Connected Program at Cheenta

#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Similar Problems

## Partial Differentiation | IIT JAM 2017 | Problem 5

Try this problem from IIT JAM 2017 exam (Problem 5).It deals with calculating the partial derivative of a multi-variable function.

## Rolle’s Theorem | IIT JAM 2017 | Problem 10

Try this problem from IIT JAM 2017 exam (Problem 10).You will need the concept of Rolle’s Theorem to solve it. You can use the sequential hints.

## Radius of Convergence of a Power series | IIT JAM 2016

Try this problem from IIT JAM 2017 exam (Problem 48) and know how to determine radius of convergence of a power series.We provide sequential Hints.

## Eigen Value of a matrix | IIT JAM 2017 | Problem 58

Try this problem from IIT JAM 2017 exam (Problem 58) and know how to evaluate Eigen value of a Matrix. We provide sequential hints.

## Limit of a function | IIT JAM 2017 | Problem 8

Try this problem from IIT JAM 2017 exam (Problem 8). It deals with evaluating Limit of a function. We provide sequential hints.

## Gradient, Divergence and Curl | IIT JAM 2014 | Problem 5

Try this problem from IIT JAM 2014 exam. It deals with calculating Gradient of a scalar point function, Divergence and curl of a vector point function point function.. We provide sequential hints.

## Differential Equation| IIT JAM 2014 | Problem 4

Try this problem from IIT JAM 2014 exam. It requires knowledge of exact differential equation and partial derivative. We provide sequential hints.

## Definite Integral as Limit of a sum | ISI QMS | QMA 2019

Try this problem from ISI QMS 2019 exam. It requires knowledge Real Analysis and integral calculus and is based on Definite Integral as Limit of a sum.

## Minimal Polynomial of a Matrix | TIFR GS-2018 (Part B)

Try this beautiful problem from TIFR GS 2018 (Part B) based on Minimal Polynomial of a Matrix. This problem requires knowledge linear algebra.

## Definite Integral & Expansion of a Determinant |ISI QMS 2019 |QMB Problem 7(a)

Try this beautiful problem from ISI QMS 2019 exam. This problem requires knowledge of determinant and definite integral. Sequential hints are given here.