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January 12, 2020

Sequences & Subsequences : IIT 2018 Problem 10

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What are we learning?

[/et_pb_text][et_pb_text _builder_version="4.0.9" text_font="Raleway||||||||" text_font_size="18px" background_color="#f4f4f4" custom_margin="50px||50px||false|false" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Sequences & Subsequences are the key features in the filed of real analysis. We will see how to imply these concepts in our problem

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Try to answer this question

[/et_pb_text][et_pb_code _builder_version="4.0.9"][h5p id="10"][/et_pb_code][et_pb_text _builder_version="4.1" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="50px||50px||false|false" custom_padding="10px|20px|10px|20px" hover_enabled="0" box_shadow_style="preset2"]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. [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.0.9" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.0.9"]IIT Jam 2018[/et_pb_accordion_item][et_pb_accordion_item title="Key competency" _builder_version="4.0.9" open="off"]Gradient[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.0.9" open="off"]Easy[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0.9" open="off"]
Calculus: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability – Vol 2Tom M. Apostol
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Start with hints

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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?                                

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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.

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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.

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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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Look at the knowledge graph...

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Play with graph

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

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Augustin-Louis Cauchy

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Connected Program at Cheenta

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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.[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/collegeprogram/" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems

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