Mixing
On inf and sup of a sequence
My question is following
Let $(x_{n})$ be a positive sequence and satisfies the inequalities
$frac{1}{K+x_{n+1}}leq x_{n}leq frac{1}{e^{-v}+x_{n+1}}$
where $K>1$ and $v>0$. How to find inf $x_{n}$ and sup $x_{n}$?
My try: Iterating the given inequalities I have found upper and lower bounds for $x_{n}$ as follow
$$
frac{sqrt{K^{2}+4Ke^{v}}-K}{2Ke^{v}}leq x_{n} leq frac{sqrt{K^{2}+4Ke^{v}}-K}{2}.
$$
But I am not so sure whether they are inf and sup?
real-analysis sequences-and-series inequality
asked Nov 20 '18 at 12:24
Alex
112
add a comment |
My question is following
Let $(x_{n})$ be a positive sequence and satisfies the inequalities
$frac{1}{K+x_{n+1}}leq x_{n}leq frac{1}{e^{-v}+x_{n+1}}$
where $K>1$ and $v>0$. How to find inf $x_{n}$ and sup $x_{n}$?
My try: Iterating the given inequalities I have found upper and lower bounds for $x_{n}$ as follow
$$
frac{sqrt{K^{2}+4Ke^{v}}-K}{2Ke^{v}}leq x_{n} leq frac{sqrt{K^{2}+4Ke^{v}}-K}{2}.
$$
But I am not so sure whether they are inf and sup?
real-analysis sequences-and-series inequality
asked Nov 20 '18 at 12:24
Alex
112
May I ask how the iteration is done? Edit: I suppose it's "Solving homogeneous linear recurrence relations with constant coefficients", right? Explained also in Wikipedia.
– Abdullah UYU
Nov 20 '18 at 13:02
Do you know something about $x_0$?
– Keen-ameteur
Nov 20 '18 at 13:04
@AbdullahUYU. Using given inequalities we can show $x_{n}leq frac{1}{e^{-v}+frac{1}{K+x_{n+2}}} $. Then we can continue this inequaity for infinite level
– Alex
Nov 20 '18 at 13:06
@Keen-ameteur I know only $x_{0}in (0,1)$
– Alex
Nov 20 '18 at 13:08
add a comment |
My question is following
Let $(x_{n})$ be a positive sequence and satisfies the inequalities
$frac{1}{K+x_{n+1}}leq x_{n}leq frac{1}{e^{-v}+x_{n+1}}$
where $K>1$ and $v>0$. How to find inf $x_{n}$ and sup $x_{n}$?
My try: Iterating the given inequalities I have found upper and lower bounds for $x_{n}$ as follow
$$
frac{sqrt{K^{2}+4Ke^{v}}-K}{2Ke^{v}}leq x_{n} leq frac{sqrt{K^{2}+4Ke^{v}}-K}{2}.
$$
But I am not so sure whether they are inf and sup?
real-analysis sequences-and-series inequality
asked Nov 20 '18 at 12:24
Alex
112
My question is following
Let $(x_{n})$ be a positive sequence and satisfies the inequalities
$frac{1}{K+x_{n+1}}leq x_{n}leq frac{1}{e^{-v}+x_{n+1}}$
where $K>1$ and $v>0$. How to find inf $x_{n}$ and sup $x_{n}$?
My try: Iterating the given inequalities I have found upper and lower bounds for $x_{n}$ as follow
$$
frac{sqrt{K^{2}+4Ke^{v}}-K}{2Ke^{v}}leq x_{n} leq frac{sqrt{K^{2}+4Ke^{v}}-K}{2}.
$$
But I am not so sure whether they are inf and sup?
real-analysis sequences-and-series inequality
real-analysis sequences-and-series inequality
asked Nov 20 '18 at 12:24
Alex
112
asked Nov 20 '18 at 12:24
Alex
112
edited Nov 20 '18 at 13:00
asked Nov 20 '18 at 12:24
Alex
112
asked Nov 20 '18 at 12:24
Alex
112
asked Nov 20 '18 at 12:24
Alex
112
112
May I ask how the iteration is done? Edit: I suppose it's "Solving homogeneous linear recurrence relations with constant coefficients", right? Explained also in Wikipedia.
– Abdullah UYU
Nov 20 '18 at 13:02
Do you know something about $x_0$?
– Keen-ameteur
Nov 20 '18 at 13:04
@AbdullahUYU. Using given inequalities we can show $x_{n}leq frac{1}{e^{-v}+frac{1}{K+x_{n+2}}} $. Then we can continue this inequaity for infinite level
– Alex
Nov 20 '18 at 13:06
@Keen-ameteur I know only $x_{0}in (0,1)$
– Alex
Nov 20 '18 at 13:08
add a comment |
May I ask how the iteration is done? Edit: I suppose it's "Solving homogeneous linear recurrence relations with constant coefficients", right? Explained also in Wikipedia.
– Abdullah UYU
Nov 20 '18 at 13:02
Do you know something about $x_0$?
– Keen-ameteur
Nov 20 '18 at 13:04
@AbdullahUYU. Using given inequalities we can show $x_{n}leq frac{1}{e^{-v}+frac{1}{K+x_{n+2}}} $. Then we can continue this inequaity for infinite level
– Alex
Nov 20 '18 at 13:06
@Keen-ameteur I know only $x_{0}in (0,1)$
– Alex
Nov 20 '18 at 13:08
May I ask how the iteration is done? Edit: I suppose it's "Solving homogeneous linear recurrence relations with constant coefficients", right? Explained also in Wikipedia.
– Abdullah UYU
Nov 20 '18 at 13:02
May I ask how the iteration is done? Edit: I suppose it's "Solving homogeneous linear recurrence relations with constant coefficients", right? Explained also in Wikipedia.
– Abdullah UYU
Nov 20 '18 at 13:02
Do you know something about $x_0$?
– Keen-ameteur
Nov 20 '18 at 13:04
Do you know something about $x_0$?
– Keen-ameteur
Nov 20 '18 at 13:04
@AbdullahUYU. Using given inequalities we can show $x_{n}leq frac{1}{e^{-v}+frac{1}{K+x_{n+2}}} $. Then we can continue this inequaity for infinite level
– Alex
Nov 20 '18 at 13:06
@AbdullahUYU. Using given inequalities we can show $x_{n}leq frac{1}{e^{-v}+frac{1}{K+x_{n+2}}} $. Then we can continue this inequaity for infinite level
– Alex
Nov 20 '18 at 13:06
@Keen-ameteur I know only $x_{0}in (0,1)$
– Alex
Nov 20 '18 at 13:08
@Keen-ameteur I know only $x_{0}in (0,1)$
– Alex
Nov 20 '18 at 13:08
add a comment |
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May I ask how the iteration is done? Edit: I suppose it's "Solving homogeneous linear recurrence relations with constant coefficients", right? Explained also in Wikipedia.
– Abdullah UYU
Nov 20 '18 at 13:02
Do you know something about $x_0$?
– Keen-ameteur
Nov 20 '18 at 13:04
@AbdullahUYU. Using given inequalities we can show $x_{n}leq frac{1}{e^{-v}+frac{1}{K+x_{n+2}}} $. Then we can continue this inequaity for infinite level
– Alex
Nov 20 '18 at 13:06
@Keen-ameteur I know only $x_{0}in (0,1)$
– Alex
Nov 20 '18 at 13:08