# 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

# Try to answer this question

# Understand the problem

(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.

##### Source of the problem

##### Key competency

##### Difficulty Level

##### Suggested Book

# Start with hints

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.

# Look at the knowledge graph…

# Play with graph

*Fun fact*: Do you know that this man has a sequence named after him?

# Augustin-Louis Cauchy

# Connected Program at Cheenta

#### College Mathematics Program

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