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Cauchy sequence

  • Thread starter dirk_mec1
  • Start date
  • #1
761
13

Homework Statement



Given:

[tex]x_{n+1}=\frac{1}{3+x_n}[/tex]

with
[tex] x_1=1 [/tex]

Show that:

(1)

[tex]|x_{n+1}-x_n| \leq \frac{1}{9}|x_{n}-x_{n-1}|[/tex]

and (2) x_n is Cauchy.

Homework Equations





The Attempt at a Solution


I've tried different approaches (including induction) but the sequence isn't monotonically decreasing.
 

Answers and Replies

  • #2
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
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Have you tried writing out [itex] |x_{n+1} - x_{n}| [/itex] using the definition [itex] x_j = 1/(3+x_{j-1}[/itex]?
 
  • #3
761
13
Got it ty.
 

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