Convergence of $x_{k+1} := left(1-frac{3sigma}{2 sqrt{lVert x_k rVert}}right) x_k$
In $(mathbb{R}^n,lVert cdot rVert := {lVert cdot rVert}_2)$, consider the sequence $(x_k)_{kin mathbb{N}_0}$,
$$
x_{k+1} := left(1-frac{3sigma}{2 sqrt{lVert x_k rVert}}right) x_k
$$
I want to show that this sequence does either hit $0$ at some $k$ or diverges, dependent on $sigma$ and $x_0$.
What I tried so far:
Writing
$$
lVert x_{k+1} rVert = left| 1- frac{3sigma}{2 sqrt{lVert x_k rVert}}right| cdot lVert x_k rVert
$$
and checking, when the factor in front of $lVert x_k rVert$ is greater than $1$. This is the case iff $sigma geq frac{4}{3} sqrt{lVert x_k rVert}$, however this doesn't seem to get me anywhere.
How can I continue? Or should I try completely another way?
real-analysis sequences-and-series convergence
asked Nov 27 '18 at 15:25
bruderjakob17
1587
add a comment |
In $(mathbb{R}^n,lVert cdot rVert := {lVert cdot rVert}_2)$, consider the sequence $(x_k)_{kin mathbb{N}_0}$,
$$
x_{k+1} := left(1-frac{3sigma}{2 sqrt{lVert x_k rVert}}right) x_k
$$
I want to show that this sequence does either hit $0$ at some $k$ or diverges, dependent on $sigma$ and $x_0$.
What I tried so far:
Writing
$$
lVert x_{k+1} rVert = left| 1- frac{3sigma}{2 sqrt{lVert x_k rVert}}right| cdot lVert x_k rVert
$$
and checking, when the factor in front of $lVert x_k rVert$ is greater than $1$. This is the case iff $sigma geq frac{4}{3} sqrt{lVert x_k rVert}$, however this doesn't seem to get me anywhere.
How can I continue? Or should I try completely another way?
real-analysis sequences-and-series convergence
asked Nov 27 '18 at 15:25
bruderjakob17
1587
Is $x_k$ a vector or a scalar?
– Mostafa Ayaz
Nov 28 '18 at 16:16
$x_k in mathbb{R}^n$
– bruderjakob17
Nov 28 '18 at 17:45
add a comment |
In $(mathbb{R}^n,lVert cdot rVert := {lVert cdot rVert}_2)$, consider the sequence $(x_k)_{kin mathbb{N}_0}$,
$$
x_{k+1} := left(1-frac{3sigma}{2 sqrt{lVert x_k rVert}}right) x_k
$$
I want to show that this sequence does either hit $0$ at some $k$ or diverges, dependent on $sigma$ and $x_0$.
What I tried so far:
Writing
$$
lVert x_{k+1} rVert = left| 1- frac{3sigma}{2 sqrt{lVert x_k rVert}}right| cdot lVert x_k rVert
$$
and checking, when the factor in front of $lVert x_k rVert$ is greater than $1$. This is the case iff $sigma geq frac{4}{3} sqrt{lVert x_k rVert}$, however this doesn't seem to get me anywhere.
How can I continue? Or should I try completely another way?
real-analysis sequences-and-series convergence
asked Nov 27 '18 at 15:25
bruderjakob17
1587
In $(mathbb{R}^n,lVert cdot rVert := {lVert cdot rVert}_2)$, consider the sequence $(x_k)_{kin mathbb{N}_0}$,
$$
x_{k+1} := left(1-frac{3sigma}{2 sqrt{lVert x_k rVert}}right) x_k
$$
I want to show that this sequence does either hit $0$ at some $k$ or diverges, dependent on $sigma$ and $x_0$.
What I tried so far:
Writing
$$
lVert x_{k+1} rVert = left| 1- frac{3sigma}{2 sqrt{lVert x_k rVert}}right| cdot lVert x_k rVert
$$
and checking, when the factor in front of $lVert x_k rVert$ is greater than $1$. This is the case iff $sigma geq frac{4}{3} sqrt{lVert x_k rVert}$, however this doesn't seem to get me anywhere.
How can I continue? Or should I try completely another way?
real-analysis sequences-and-series convergence
real-analysis sequences-and-series convergence
asked Nov 27 '18 at 15:25
bruderjakob17
1587
asked Nov 27 '18 at 15:25
bruderjakob17
1587
asked Nov 27 '18 at 15:25
bruderjakob17
1587
asked Nov 27 '18 at 15:25
bruderjakob17
1587
asked Nov 27 '18 at 15:25
bruderjakob17
1587
1587
Is $x_k$ a vector or a scalar?
– Mostafa Ayaz
Nov 28 '18 at 16:16
$x_k in mathbb{R}^n$
– bruderjakob17
Nov 28 '18 at 17:45
add a comment |
Is $x_k$ a vector or a scalar?
– Mostafa Ayaz
Nov 28 '18 at 16:16
$x_k in mathbb{R}^n$
– bruderjakob17
Nov 28 '18 at 17:45
Is $x_k$ a vector or a scalar?
– Mostafa Ayaz
Nov 28 '18 at 16:16
Is $x_k$ a vector or a scalar?
– Mostafa Ayaz
Nov 28 '18 at 16:16
$x_k in mathbb{R}^n$
– bruderjakob17
Nov 28 '18 at 17:45
$x_k in mathbb{R}^n$
– bruderjakob17
Nov 28 '18 at 17:45
add a comment |
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Is $x_k$ a vector or a scalar?
– Mostafa Ayaz
Nov 28 '18 at 16:16
$x_k in mathbb{R}^n$
– bruderjakob17
Nov 28 '18 at 17:45