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# 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.

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#### College Mathematics Program

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