# 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

##### Source of the problem

##### Key competency

##### Difficulty Level

# 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

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.

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