Tricky AM-GM inequalityProof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$How to use induction on this type of inequality?How to prove $a_1^m + a_2^m + cdots + a_n^m geq frac1a_1 + frac1a_2 + cdots + frac1a_n$Bernoulli's inequality variationInequality $( sum_i=1^n a_i)( sum_i=1^n frac1a_i)geq n^2$Proof of this inequalityRearrangement Inequality Problem: Mathematical Olympiad.General inequalityInequality : $ (a_1a_2+a_2a_3+ldots+a_na_1)left(fraca_1a^2_2+a_2+fraca_2a^2_3+a_3+ ldots+fraca_na^2_1+a_1right)geq fracnn+1 $Regarding AM-GM inequalityInequality with two sums
How much stiffer are 23c tires over 28c?
PL tone removal?
Why is there a voltage between the mains ground and my radiator?
Given the sum of two powers of two, extract the exponents
Why do different render engines generate different z pass?
Unreachable code, but reachable with exception
Algorithm to convert a fixed-length string to the smallest possible collision-free representation?
Can a bounded number sequence be strictly ascending?
A three room house but a three headED dog
How to pass a string to a command that expects a file?
How do you like my writing?
htop displays identical program in multiple lines
GPLv2 - licensing for commercial use
Do f-stop and exposure time perfectly cancel?
What to do when during a meeting client people start to fight (even physically) with each others?
Aliens englobed the Solar System: will we notice?
How did the power source of Mar-Vell's aircraft end up with her?
Should I tell my boss the work he did was worthless
Why does Deadpool say "You're welcome, Canada," after shooting Ryan Reynolds in the end credits?
Making a sword in the stone, in a medieval world without magic
Should I take out a loan for a friend to invest on my behalf?
If the Captain's screens are out, does he switch seats with the co-pilot?
Offered promotion but I'm leaving. Should I tell?
How much attack damage does the AC boost from a shield prevent on average?
Tricky AM-GM inequality
Proof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$How to use induction on this type of inequality?How to prove $a_1^m + a_2^m + cdots + a_n^m geq frac1a_1 + frac1a_2 + cdots + frac1a_n$Bernoulli's inequality variationInequality $( sum_i=1^n a_i)( sum_i=1^n frac1a_i)geq n^2$Proof of this inequalityRearrangement Inequality Problem: Mathematical Olympiad.General inequalityInequality : $ (a_1a_2+a_2a_3+ldots+a_na_1)left(fraca_1a^2_2+a_2+fraca_2a^2_3+a_3+ ldots+fraca_na^2_1+a_1right)geq fracnn+1 $Regarding AM-GM inequalityInequality with two sums
$begingroup$
I've been struggling for several hours, trying to prove this horrible inequality:
$(a_1+a_2+dotsb+a_n)left(frac1a_1+frac1a_2+dotsb+frac1a_nright)geq n^2$.
Where each $a_i$'s are positive and $n$ is a natural number.
First I tried the usual "mathematical induction" method, but it made no avail, since I could not show it would be true if n=k+1.
Suppose the inequality holds true when n=k, i.e.,
$(a_1+a_2+dotsb+a_k)left(frac1a_1+frac1a_2+dotsb+frac1a_kright)geq n^2$.
This is true if and only if
$(a_1+a_2+dotsb+a_k+a_k+1)left(frac1a_1+frac1a_2+dotsb+frac1a_k+frac1a_k+1right) -a_k+1left(frac1a_1+dotsb+frac1a_kright)-frac1a_k+1(a_1+dotsb+a_k)-fraca_k+1a_k+1 geq n^2$.
And I got stuck here.
The question looks like I have to use AM-GM inequality at some point, but I do not have a clue. Any small hints and clues will be appreciated.
analysis inequality
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
$endgroup$
add a comment |
$begingroup$
I've been struggling for several hours, trying to prove this horrible inequality:
$(a_1+a_2+dotsb+a_n)left(frac1a_1+frac1a_2+dotsb+frac1a_nright)geq n^2$.
Where each $a_i$'s are positive and $n$ is a natural number.
First I tried the usual "mathematical induction" method, but it made no avail, since I could not show it would be true if n=k+1.
Suppose the inequality holds true when n=k, i.e.,
$(a_1+a_2+dotsb+a_k)left(frac1a_1+frac1a_2+dotsb+frac1a_kright)geq n^2$.
This is true if and only if
$(a_1+a_2+dotsb+a_k+a_k+1)left(frac1a_1+frac1a_2+dotsb+frac1a_k+frac1a_k+1right) -a_k+1left(frac1a_1+dotsb+frac1a_kright)-frac1a_k+1(a_1+dotsb+a_k)-fraca_k+1a_k+1 geq n^2$.
And I got stuck here.
The question looks like I have to use AM-GM inequality at some point, but I do not have a clue. Any small hints and clues will be appreciated.
analysis inequality
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
$endgroup$
$begingroup$
Conditions on $a_i$?
$endgroup$
– Parcly Taxel
4 hours ago
$begingroup$
whoa, I forgot the most important info there. They are all positive, and n is a natural number.
$endgroup$
– Ko Byeongmin
4 hours ago
4
$begingroup$
Try Cauchy-Schwarz inequality?
$endgroup$
– Sik Feng Cheong
4 hours ago
$begingroup$
Now I get it, I learn something new every day!! Thanks a lot :D
$endgroup$
– Ko Byeongmin
4 hours ago
2
$begingroup$
Possible duplicate of Proof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$
$endgroup$
– Arnaud D.
3 hours ago
add a comment |
$begingroup$
I've been struggling for several hours, trying to prove this horrible inequality:
$(a_1+a_2+dotsb+a_n)left(frac1a_1+frac1a_2+dotsb+frac1a_nright)geq n^2$.
Where each $a_i$'s are positive and $n$ is a natural number.
First I tried the usual "mathematical induction" method, but it made no avail, since I could not show it would be true if n=k+1.
Suppose the inequality holds true when n=k, i.e.,
$(a_1+a_2+dotsb+a_k)left(frac1a_1+frac1a_2+dotsb+frac1a_kright)geq n^2$.
This is true if and only if
$(a_1+a_2+dotsb+a_k+a_k+1)left(frac1a_1+frac1a_2+dotsb+frac1a_k+frac1a_k+1right) -a_k+1left(frac1a_1+dotsb+frac1a_kright)-frac1a_k+1(a_1+dotsb+a_k)-fraca_k+1a_k+1 geq n^2$.
And I got stuck here.
The question looks like I have to use AM-GM inequality at some point, but I do not have a clue. Any small hints and clues will be appreciated.
analysis inequality
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
$endgroup$
I've been struggling for several hours, trying to prove this horrible inequality:
$(a_1+a_2+dotsb+a_n)left(frac1a_1+frac1a_2+dotsb+frac1a_nright)geq n^2$.
Where each $a_i$'s are positive and $n$ is a natural number.
First I tried the usual "mathematical induction" method, but it made no avail, since I could not show it would be true if n=k+1.
Suppose the inequality holds true when n=k, i.e.,
$(a_1+a_2+dotsb+a_k)left(frac1a_1+frac1a_2+dotsb+frac1a_kright)geq n^2$.
This is true if and only if
$(a_1+a_2+dotsb+a_k+a_k+1)left(frac1a_1+frac1a_2+dotsb+frac1a_k+frac1a_k+1right) -a_k+1left(frac1a_1+dotsb+frac1a_kright)-frac1a_k+1(a_1+dotsb+a_k)-fraca_k+1a_k+1 geq n^2$.
And I got stuck here.
The question looks like I have to use AM-GM inequality at some point, but I do not have a clue. Any small hints and clues will be appreciated.
analysis inequality
analysis inequality
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
edited 4 hours ago
Ko Byeongmin
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
asked 4 hours ago
Ko ByeongminKo Byeongmin
1546
1546
$begingroup$
Conditions on $a_i$?
$endgroup$
– Parcly Taxel
4 hours ago
$begingroup$
whoa, I forgot the most important info there. They are all positive, and n is a natural number.
$endgroup$
– Ko Byeongmin
4 hours ago
4
$begingroup$
Try Cauchy-Schwarz inequality?
$endgroup$
– Sik Feng Cheong
4 hours ago
$begingroup$
Now I get it, I learn something new every day!! Thanks a lot :D
$endgroup$
– Ko Byeongmin
4 hours ago
2
$begingroup$
Possible duplicate of Proof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$
$endgroup$
– Arnaud D.
3 hours ago
add a comment |
$begingroup$
Conditions on $a_i$?
$endgroup$
– Parcly Taxel
4 hours ago
$begingroup$
whoa, I forgot the most important info there. They are all positive, and n is a natural number.
$endgroup$
– Ko Byeongmin
4 hours ago
4
$begingroup$
Try Cauchy-Schwarz inequality?
$endgroup$
– Sik Feng Cheong
4 hours ago
$begingroup$
Now I get it, I learn something new every day!! Thanks a lot :D
$endgroup$
– Ko Byeongmin
4 hours ago
2
$begingroup$
Possible duplicate of Proof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$
$endgroup$
– Arnaud D.
3 hours ago
$begingroup$
Conditions on $a_i$?
$endgroup$
– Parcly Taxel
4 hours ago
$begingroup$
Conditions on $a_i$?
$endgroup$
– Parcly Taxel
4 hours ago
$begingroup$
whoa, I forgot the most important info there. They are all positive, and n is a natural number.
$endgroup$
– Ko Byeongmin
4 hours ago
$begingroup$
whoa, I forgot the most important info there. They are all positive, and n is a natural number.
$endgroup$
– Ko Byeongmin
4 hours ago
4
4
$begingroup$
Try Cauchy-Schwarz inequality?
$endgroup$
– Sik Feng Cheong
4 hours ago
$begingroup$
Try Cauchy-Schwarz inequality?
$endgroup$
– Sik Feng Cheong
4 hours ago
$begingroup$
Now I get it, I learn something new every day!! Thanks a lot :D
$endgroup$
– Ko Byeongmin
4 hours ago
$begingroup$
Now I get it, I learn something new every day!! Thanks a lot :D
$endgroup$
– Ko Byeongmin
4 hours ago
2
2
$begingroup$
Possible duplicate of Proof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$
$endgroup$
– Arnaud D.
3 hours ago
$begingroup$
Possible duplicate of Proof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$
$endgroup$
– Arnaud D.
3 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint: AM-GM implies
$$
a_1+a_2+cdots +a_nge nsqrt[n]a_1a_2cdots a_n
$$ and $$
frac1a_1+frac1a_2+cdots +frac1a_nge fracnsqrt[n]a_1a_2cdots a_n.
$$
answered 4 hours ago
SongSong
17.2k21246
$endgroup$
$begingroup$
That's a really strong hint, it looks like an answer
$endgroup$
– enedil
2 hours ago
$begingroup$
You may be right.. It makes the remaining step look so trivial, so it can be justly regarded as an answer.
$endgroup$
– Song
2 hours ago
add a comment |
$begingroup$
It is AM-HM inequality
$$fraca_1+a_2+a_3+...+a_nngeq fracnfrac1a_1+frac1a_2+frac1a_3+...+frac1a_n$$
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
$endgroup$
add a comment |
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',
autoActivateHeartbeat: false,
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%2f3145187%2ftricky-am-gm-inequality%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: AM-GM implies
$$
a_1+a_2+cdots +a_nge nsqrt[n]a_1a_2cdots a_n
$$ and $$
frac1a_1+frac1a_2+cdots +frac1a_nge fracnsqrt[n]a_1a_2cdots a_n.
$$
answered 4 hours ago
SongSong
17.2k21246
$endgroup$
$begingroup$
That's a really strong hint, it looks like an answer
$endgroup$
– enedil
2 hours ago
$begingroup$
You may be right.. It makes the remaining step look so trivial, so it can be justly regarded as an answer.
$endgroup$
– Song
2 hours ago
add a comment |
$begingroup$
Hint: AM-GM implies
$$
a_1+a_2+cdots +a_nge nsqrt[n]a_1a_2cdots a_n
$$ and $$
frac1a_1+frac1a_2+cdots +frac1a_nge fracnsqrt[n]a_1a_2cdots a_n.
$$
answered 4 hours ago
SongSong
17.2k21246
$endgroup$
$begingroup$
That's a really strong hint, it looks like an answer
$endgroup$
– enedil
2 hours ago
$begingroup$
You may be right.. It makes the remaining step look so trivial, so it can be justly regarded as an answer.
$endgroup$
– Song
2 hours ago
add a comment |
$begingroup$
Hint: AM-GM implies
$$
a_1+a_2+cdots +a_nge nsqrt[n]a_1a_2cdots a_n
$$ and $$
frac1a_1+frac1a_2+cdots +frac1a_nge fracnsqrt[n]a_1a_2cdots a_n.
$$
answered 4 hours ago
SongSong
17.2k21246
$endgroup$
Hint: AM-GM implies
$$
a_1+a_2+cdots +a_nge nsqrt[n]a_1a_2cdots a_n
$$ and $$
frac1a_1+frac1a_2+cdots +frac1a_nge fracnsqrt[n]a_1a_2cdots a_n.
$$
answered 4 hours ago
SongSong
17.2k21246
answered 4 hours ago
SongSong
17.2k21246
answered 4 hours ago
SongSong
17.2k21246
answered 4 hours ago
SongSong
17.2k21246
17.2k21246
$begingroup$
That's a really strong hint, it looks like an answer
$endgroup$
– enedil
2 hours ago
$begingroup$
You may be right.. It makes the remaining step look so trivial, so it can be justly regarded as an answer.
$endgroup$
– Song
2 hours ago
add a comment |
$begingroup$
That's a really strong hint, it looks like an answer
$endgroup$
– enedil
2 hours ago
$begingroup$
You may be right.. It makes the remaining step look so trivial, so it can be justly regarded as an answer.
$endgroup$
– Song
2 hours ago
$begingroup$
That's a really strong hint, it looks like an answer
$endgroup$
– enedil
2 hours ago
$begingroup$
That's a really strong hint, it looks like an answer
$endgroup$
– enedil
2 hours ago
$begingroup$
You may be right.. It makes the remaining step look so trivial, so it can be justly regarded as an answer.
$endgroup$
– Song
2 hours ago
$begingroup$
You may be right.. It makes the remaining step look so trivial, so it can be justly regarded as an answer.
$endgroup$
– Song
2 hours ago
add a comment |
$begingroup$
It is AM-HM inequality
$$fraca_1+a_2+a_3+...+a_nngeq fracnfrac1a_1+frac1a_2+frac1a_3+...+frac1a_n$$
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
$endgroup$
add a comment |
$begingroup$
It is AM-HM inequality
$$fraca_1+a_2+a_3+...+a_nngeq fracnfrac1a_1+frac1a_2+frac1a_3+...+frac1a_n$$
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
$endgroup$
add a comment |
$begingroup$
It is AM-HM inequality
$$fraca_1+a_2+a_3+...+a_nngeq fracnfrac1a_1+frac1a_2+frac1a_3+...+frac1a_n$$
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
$endgroup$
It is AM-HM inequality
$$fraca_1+a_2+a_3+...+a_nngeq fracnfrac1a_1+frac1a_2+frac1a_3+...+frac1a_n$$
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
answered 4 hours ago
Dr. Sonnhard GraubnerDr. Sonnhard Graubner
77.7k42866
77.7k42866
add a comment |
add a comment |
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.
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%2f3145187%2ftricky-am-gm-inequality%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
-analysis, inequality
$begingroup$
Conditions on $a_i$?
$endgroup$
– Parcly Taxel
4 hours ago
$begingroup$
whoa, I forgot the most important info there. They are all positive, and n is a natural number.
$endgroup$
– Ko Byeongmin
4 hours ago
4
$begingroup$
Try Cauchy-Schwarz inequality?
$endgroup$
– Sik Feng Cheong
4 hours ago
$begingroup$
Now I get it, I learn something new every day!! Thanks a lot :D
$endgroup$
– Ko Byeongmin
4 hours ago
2
$begingroup$
Possible duplicate of Proof that $left(sum^n_k=1x_kright)left(sum^n_k=1y_kright)geq n^2$
$endgroup$
– Arnaud D.
3 hours ago