$| U_{n+1} - 2/3|leqslant 1/2 | U_n-2/3 |+ epsilon$
up vote
0
down vote
favorite
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 at 21:06
Daniel
1,516210
asked Nov 21 at 20:03
Alae Cherkaoui
84
|
show 2 more comments
up vote
0
down vote
favorite
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 at 21:06
Daniel
1,516210
asked Nov 21 at 20:03
Alae Cherkaoui
84
This question makes no sense
– Daniel
Nov 21 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 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 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 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 at 21:29
|
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 at 21:06
Daniel
1,516210
asked Nov 21 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 at 21:06
Daniel
1,516210
asked Nov 21 at 20:03
Alae Cherkaoui
84
edited Nov 21 at 21:06
Daniel
1,516210
asked Nov 21 at 20:03
Alae Cherkaoui
84
edited Nov 21 at 21:06
Daniel
1,516210
edited Nov 21 at 21:06
Daniel
1,516210
edited Nov 21 at 21:06
Daniel
1,516210
1,516210
asked Nov 21 at 20:03
Alae Cherkaoui
84
asked Nov 21 at 20:03
Alae Cherkaoui
84
asked Nov 21 at 20:03
Alae Cherkaoui
84
84
This question makes no sense
– Daniel
Nov 21 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 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 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 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 at 21:29
|
show 2 more comments
This question makes no sense
– Daniel
Nov 21 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 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 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 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 at 21:29
This question makes no sense
– Daniel
Nov 21 at 20:10
This question makes no sense
– Daniel
Nov 21 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 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 at 20:18
2
2
You did not give the definition of $U_n$
– Daniel
Nov 21 at 20:23
You did not give the definition of $U_n$
– Daniel
Nov 21 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 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 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 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 at 21:29
|
show 2 more comments
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008270%2fu-n1-2-3-leqslant-1-2-u-n-2-3-epsilon%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008270%2fu-n1-2-3-leqslant-1-2-u-n-2-3-epsilon%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
This question makes no sense
– Daniel
Nov 21 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 at 20:18
2
You did not give the definition of $U_n$
– Daniel
Nov 21 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 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 at 21:29