$| U_{n+1} - 2/3|leqslant 1/2 | U_n-2/3 |+ epsilon$
I'm not able to find a way to prove this inequality:
"$U_n$ is defined by
$U_0 = U_1=3/2$ and $U_{n+2}=1 + sqrt{U_{n+1}U_n}$.
Fix a real number '$epsilon$' that is strictly positive.
Show that there is a natural number $n_epsilon$ such that for
every $n$ bigger than $n_epsilon$, $| U_{n+1} - 2/3| leqslant 1/2 |U_n-2/3| + epsilon$."
Any help regarding this problem is appreciated. Thanks in advance.
sequences-and-series
edited Nov 21 '18 at 21:06
Daniel
1,516210
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
|
show 2 more comments
I'm not able to find a way to prove this inequality:
"$U_n$ is defined by
$U_0 = U_1=3/2$ and $U_{n+2}=1 + sqrt{U_{n+1}U_n}$.
Fix a real number '$epsilon$' that is strictly positive.
Show that there is a natural number $n_epsilon$ such that for
every $n$ bigger than $n_epsilon$, $| U_{n+1} - 2/3| leqslant 1/2 |U_n-2/3| + epsilon$."
Any help regarding this problem is appreciated. Thanks in advance.
sequences-and-series
edited Nov 21 '18 at 21:06
Daniel
1,516210
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
This question makes no sense
– Daniel
Nov 21 '18 at 20:10
@Daniel link You can check it on Exercice 7 , It's french but I can translate for you.
– Alae Cherkaoui
Nov 21 '18 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 '18 at 20:23
1
Notice that in the picture you sent, the exercise 6 also mentions sequences $v_n$ and $u_n$ that seem to be already defined. Look the previous exercises and find the definition of $U_n$ (actually, $v_n$ in the book), otherwise it makes no sense.
– Daniel
Nov 21 '18 at 20:56
1
You should try to do the previous exercises, since it seems they will help you with this one. Applying the triangular inequality to the equation on exercise 6 gives $|v_{n+1} - frac23| le mid frac12(v_n - frac23)mid + |v_n alpha_n| $ and apparently this $alpha_n$ converges to $0$...
– Daniel
Nov 21 '18 at 21:29
|
show 2 more comments
I'm not able to find a way to prove this inequality:
"$U_n$ is defined by
$U_0 = U_1=3/2$ and $U_{n+2}=1 + sqrt{U_{n+1}U_n}$.
Fix a real number '$epsilon$' that is strictly positive.
Show that there is a natural number $n_epsilon$ such that for
every $n$ bigger than $n_epsilon$, $| U_{n+1} - 2/3| leqslant 1/2 |U_n-2/3| + epsilon$."
Any help regarding this problem is appreciated. Thanks in advance.
sequences-and-series
edited Nov 21 '18 at 21:06
Daniel
1,516210
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
I'm not able to find a way to prove this inequality:
"$U_n$ is defined by
$U_0 = U_1=3/2$ and $U_{n+2}=1 + sqrt{U_{n+1}U_n}$.
Fix a real number '$epsilon$' that is strictly positive.
Show that there is a natural number $n_epsilon$ such that for
every $n$ bigger than $n_epsilon$, $| U_{n+1} - 2/3| leqslant 1/2 |U_n-2/3| + epsilon$."
Any help regarding this problem is appreciated. Thanks in advance.
sequences-and-series
sequences-and-series
edited Nov 21 '18 at 21:06
Daniel
1,516210
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
edited Nov 21 '18 at 21:06
Daniel
1,516210
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
edited Nov 21 '18 at 21:06
Daniel
1,516210
edited Nov 21 '18 at 21:06
Daniel
1,516210
edited Nov 21 '18 at 21:06
Daniel
1,516210
1,516210
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
asked Nov 21 '18 at 20:03
Alae Cherkaoui
84
84
This question makes no sense
– Daniel
Nov 21 '18 at 20:10
@Daniel link You can check it on Exercice 7 , It's french but I can translate for you.
– Alae Cherkaoui
Nov 21 '18 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 '18 at 20:23
1
Notice that in the picture you sent, the exercise 6 also mentions sequences $v_n$ and $u_n$ that seem to be already defined. Look the previous exercises and find the definition of $U_n$ (actually, $v_n$ in the book), otherwise it makes no sense.
– Daniel
Nov 21 '18 at 20:56
1
You should try to do the previous exercises, since it seems they will help you with this one. Applying the triangular inequality to the equation on exercise 6 gives $|v_{n+1} - frac23| le mid frac12(v_n - frac23)mid + |v_n alpha_n| $ and apparently this $alpha_n$ converges to $0$...
– Daniel
Nov 21 '18 at 21:29
|
show 2 more comments
This question makes no sense
– Daniel
Nov 21 '18 at 20:10
@Daniel link You can check it on Exercice 7 , It's french but I can translate for you.
– Alae Cherkaoui
Nov 21 '18 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 '18 at 20:23
1
Notice that in the picture you sent, the exercise 6 also mentions sequences $v_n$ and $u_n$ that seem to be already defined. Look the previous exercises and find the definition of $U_n$ (actually, $v_n$ in the book), otherwise it makes no sense.
– Daniel
Nov 21 '18 at 20:56
1
You should try to do the previous exercises, since it seems they will help you with this one. Applying the triangular inequality to the equation on exercise 6 gives $|v_{n+1} - frac23| le mid frac12(v_n - frac23)mid + |v_n alpha_n| $ and apparently this $alpha_n$ converges to $0$...
– Daniel
Nov 21 '18 at 21:29
This question makes no sense
– Daniel
Nov 21 '18 at 20:10
This question makes no sense
– Daniel
Nov 21 '18 at 20:10
@Daniel link You can check it on Exercice 7 , It's french but I can translate for you.
– Alae Cherkaoui
Nov 21 '18 at 20:18
@Daniel link You can check it on Exercice 7 , It's french but I can translate for you.
– Alae Cherkaoui
Nov 21 '18 at 20:18
2
2
You did not give the definition of $U_n$
– Daniel
Nov 21 '18 at 20:23
You did not give the definition of $U_n$
– Daniel
Nov 21 '18 at 20:23
1
1
Notice that in the picture you sent, the exercise 6 also mentions sequences $v_n$ and $u_n$ that seem to be already defined. Look the previous exercises and find the definition of $U_n$ (actually, $v_n$ in the book), otherwise it makes no sense.
– Daniel
Nov 21 '18 at 20:56
Notice that in the picture you sent, the exercise 6 also mentions sequences $v_n$ and $u_n$ that seem to be already defined. Look the previous exercises and find the definition of $U_n$ (actually, $v_n$ in the book), otherwise it makes no sense.
– Daniel
Nov 21 '18 at 20:56
1
1
You should try to do the previous exercises, since it seems they will help you with this one. Applying the triangular inequality to the equation on exercise 6 gives $|v_{n+1} - frac23| le mid frac12(v_n - frac23)mid + |v_n alpha_n| $ and apparently this $alpha_n$ converges to $0$...
– Daniel
Nov 21 '18 at 21:29
You should try to do the previous exercises, since it seems they will help you with this one. Applying the triangular inequality to the equation on exercise 6 gives $|v_{n+1} - frac23| le mid frac12(v_n - frac23)mid + |v_n alpha_n| $ and apparently this $alpha_n$ converges to $0$...
– Daniel
Nov 21 '18 at 21:29
|
show 2 more comments
0
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This question makes no sense
– Daniel
Nov 21 '18 at 20:10
@Daniel link You can check it on Exercice 7 , It's french but I can translate for you.
– Alae Cherkaoui
Nov 21 '18 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 '18 at 20:23
1
Notice that in the picture you sent, the exercise 6 also mentions sequences $v_n$ and $u_n$ that seem to be already defined. Look the previous exercises and find the definition of $U_n$ (actually, $v_n$ in the book), otherwise it makes no sense.
– Daniel
Nov 21 '18 at 20:56
1
You should try to do the previous exercises, since it seems they will help you with this one. Applying the triangular inequality to the equation on exercise 6 gives $|v_{n+1} - frac23| le mid frac12(v_n - frac23)mid + |v_n alpha_n| $ and apparently this $alpha_n$ converges to $0$...
– Daniel
Nov 21 '18 at 21:29