O let a,=!generakd as follous:Considen thesequeuce {anyニ+a ntl= I+ IItan(a) Write down expresion fer az, as, a4> as ,as6) Show thatLim an= S2the fellomi ng(c) Using ca) ,c6) conclude that it givescontinued fraction:Sz =I+..Cd) BONUS) what happens if a;=2or a,=3, or in%3Dgeneralif a,= R ? How does the limit change?%3D

Name: O let a,=!generakd as follous:Considen thesequeuce {anyニ+a ntl= I+ II
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Description: O let a,=!generakd as follous:Considen thesequeuce {anyニ+a ntl= I+ IItan(a) Write down expresion fer az, as, a4> as ,as6) Show thatLim an= S2the fellomi ng(c) Using ca) ,c6) conclude that it givescontinued fraction:Sz =I+..Cd) BONUS) what happens if

Answered: let a,=!. Consider the sequence dany…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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O let a,=!
generakd as follous:
Considen the
sequeuce {any
ニ+
a ntl
= I+ I
Itan
(a) Write down expresion fer az, as, a4> as ,as
6) Show that
Lim an= S2
the fellomi ng
(c) Using ca) ,c6) conclude that it gives
continued fraction:
Sz =
I+
..
Cd) BONUS) what happens if a;=2
or a,=3, or in
%3D
general
if a,= R ? How does the limit change?
%3D

Expert Solution

Step 1

We will answer the first two questions since we don't answer multiple questions at a time. Please resubmit the other questions.

Step 2

The given expression is:

$a_{n + 1} = 1 + \frac{1}{1 + a_{n}}$

Step 3

So,

$\begin{array}{rcl} a_{2} & = & 1 + \frac{1}{1 + a_{1}} \\ = & 1 + \frac{1}{1 + 1} \\ = & 1 + \frac{1}{2} \\ = & \frac{3}{2} \end{array}$

$\begin{array}{rcl} a_{3} & = & 1 + \frac{1}{1 + a_{2}} \\ = & 1 + \frac{1}{1 + \frac{3}{2}} \\ = & 1 + \frac{2}{5} \\ = & \frac{7}{5} \end{array}$

$\begin{array}{rcl} a_{4} & = & 1 + \frac{1}{1 + a_{3}} \\ = & 1 + \frac{1}{1 + \frac{7}{5}} \\ = & 1 + \frac{5}{12} \\ = & \frac{17}{12} \end{array}$

Step 4

$\begin{array}{rcl} a_{5} & = & 1 + \frac{1}{1 + a_{4}} \\ = & 1 + \frac{1}{1 + \frac{17}{12}} \\ = & 1 + \frac{12}{29} \\ = & \frac{41}{29} \end{array}$

$\begin{array}{rcl} a_{6} & = & 1 + \frac{1}{1 + a_{5}} \\ = & 1 + \frac{1}{1 + \frac{41}{29}} \\ = & 1 + \frac{29}{70} \\ = & \frac{99}{70} \end{array}$

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