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Learn More**Sequences**, **Subsequences** are the key features in the field of real analysis. We will see how to imply these concepts in our 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. Difficulty Level Easy Suggested Book

Calculus: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability – Vol 2 | Tom M. Apostol |

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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