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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
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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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