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prove the inequality using inequalities like AM GM HM OR CAUCHY or WEIRSTRASS ETC.
Demonstration of Cauchy-Schwarz inequality using Minkovski inequalityHow do I prove this set of inequalities using Cauchy-Schwarz?How to prove the following using Cauchy Schwarz inequality.How to prove this inequality using Cauchy-Schwarz inequalityProve an inequality using Cauchy-Schwarz.Proving an inequality using Cauchy-SchwarzHow to solve this problem using Cauchy-Schwarz inequalityProve $x_1x_2…x_n ge (n-1)^n$Proving a Cauchy-Schwarz-like inequalityCauchy-Schwarz Inequality troubles
$begingroup$
The inequality to be proven is
$$ 2^n gt 1 + ncdot sqrt2^n-1 for all n>2 $$
using any inequalities like am gm hm cauchy schwarz tchebychev etc
I recently studied inequalities came across this question please help
inequality a.m.-g.m.-inequality cauchy-schwarz-inequality
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
The inequality to be proven is
$$ 2^n gt 1 + ncdot sqrt2^n-1 for all n>2 $$
using any inequalities like am gm hm cauchy schwarz tchebychev etc
I recently studied inequalities came across this question please help
inequality a.m.-g.m.-inequality cauchy-schwarz-inequality
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Does this look like a tight inequality for large $n$?
$endgroup$
– Mark Bennet
Mar 13 at 18:37
$begingroup$
Welcome to Mathematics Stack Exchange! Your post is going to attract downvotes with the upper case sentences. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
Mar 13 at 18:41
1
$begingroup$
Write $2^n-1$ as a sum of powers of two.
$endgroup$
– Mindlack
Mar 13 at 18:47
$begingroup$
Is n an integer?
$endgroup$
– Prakhar Neema
Mar 13 at 18:51
$begingroup$
i don't know what n is as it is not given in the question i think n is a real number.
$endgroup$
– Rishi Dev Parmar
Mar 13 at 18:56
add a comment |
$begingroup$
The inequality to be proven is
$$ 2^n gt 1 + ncdot sqrt2^n-1 for all n>2 $$
using any inequalities like am gm hm cauchy schwarz tchebychev etc
I recently studied inequalities came across this question please help
inequality a.m.-g.m.-inequality cauchy-schwarz-inequality
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
The inequality to be proven is
$$ 2^n gt 1 + ncdot sqrt2^n-1 for all n>2 $$
using any inequalities like am gm hm cauchy schwarz tchebychev etc
I recently studied inequalities came across this question please help
inequality a.m.-g.m.-inequality cauchy-schwarz-inequality
inequality a.m.-g.m.-inequality cauchy-schwarz-inequality
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Mar 13 at 18:50
Rishi Dev Parmar
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
asked Mar 13 at 18:33
Rishi Dev ParmarRishi Dev Parmar
34
34
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Rishi Dev Parmar is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Does this look like a tight inequality for large $n$?
$endgroup$
– Mark Bennet
Mar 13 at 18:37
$begingroup$
Welcome to Mathematics Stack Exchange! Your post is going to attract downvotes with the upper case sentences. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
Mar 13 at 18:41
1
$begingroup$
Write $2^n-1$ as a sum of powers of two.
$endgroup$
– Mindlack
Mar 13 at 18:47
$begingroup$
Is n an integer?
$endgroup$
– Prakhar Neema
Mar 13 at 18:51
$begingroup$
i don't know what n is as it is not given in the question i think n is a real number.
$endgroup$
– Rishi Dev Parmar
Mar 13 at 18:56
add a comment |
$begingroup$
Does this look like a tight inequality for large $n$?
$endgroup$
– Mark Bennet
Mar 13 at 18:37
$begingroup$
Welcome to Mathematics Stack Exchange! Your post is going to attract downvotes with the upper case sentences. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
Mar 13 at 18:41
1
$begingroup$
Write $2^n-1$ as a sum of powers of two.
$endgroup$
– Mindlack
Mar 13 at 18:47
$begingroup$
Is n an integer?
$endgroup$
– Prakhar Neema
Mar 13 at 18:51
$begingroup$
i don't know what n is as it is not given in the question i think n is a real number.
$endgroup$
– Rishi Dev Parmar
Mar 13 at 18:56
$begingroup$
Does this look like a tight inequality for large $n$?
$endgroup$
– Mark Bennet
Mar 13 at 18:37
$begingroup$
Does this look like a tight inequality for large $n$?
$endgroup$
– Mark Bennet
Mar 13 at 18:37
$begingroup$
Welcome to Mathematics Stack Exchange! Your post is going to attract downvotes with the upper case sentences. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
Mar 13 at 18:41
$begingroup$
Welcome to Mathematics Stack Exchange! Your post is going to attract downvotes with the upper case sentences. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
Mar 13 at 18:41
1
1
$begingroup$
Write $2^n-1$ as a sum of powers of two.
$endgroup$
– Mindlack
Mar 13 at 18:47
$begingroup$
Write $2^n-1$ as a sum of powers of two.
$endgroup$
– Mindlack
Mar 13 at 18:47
$begingroup$
Is n an integer?
$endgroup$
– Prakhar Neema
Mar 13 at 18:51
$begingroup$
Is n an integer?
$endgroup$
– Prakhar Neema
Mar 13 at 18:51
$begingroup$
i don't know what n is as it is not given in the question i think n is a real number.
$endgroup$
– Rishi Dev Parmar
Mar 13 at 18:56
$begingroup$
i don't know what n is as it is not given in the question i think n is a real number.
$endgroup$
– Rishi Dev Parmar
Mar 13 at 18:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If n $epsilon mathbbN$, apply AM-GM on $1, 2, 2^2......., 2^n-1$. $$frac1 + 2 + 2^2 +......+2^n-1n geqslant sqrt[n][1*2*....*2^n-1]$$
$$implies frac2^n - 1n geqslant sqrt[n][2^n*(n-1)/2]$$
$$implies frac2^n - 1n geqslant sqrt[2][2^n-1]$$
$$implies 2^n geqslant 1 + n*sqrt[2][2^n-1]$$
Equality holds at n=1.
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
thank you so much Prakhar Neema
$endgroup$
– Rishi Dev Parmar
Mar 13 at 19:13
add a comment |
Your Answer
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1 Answer
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1 Answer
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oldest
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votes
$begingroup$
If n $epsilon mathbbN$, apply AM-GM on $1, 2, 2^2......., 2^n-1$. $$frac1 + 2 + 2^2 +......+2^n-1n geqslant sqrt[n][1*2*....*2^n-1]$$
$$implies frac2^n - 1n geqslant sqrt[n][2^n*(n-1)/2]$$
$$implies frac2^n - 1n geqslant sqrt[2][2^n-1]$$
$$implies 2^n geqslant 1 + n*sqrt[2][2^n-1]$$
Equality holds at n=1.
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
thank you so much Prakhar Neema
$endgroup$
– Rishi Dev Parmar
Mar 13 at 19:13
add a comment |
$begingroup$
If n $epsilon mathbbN$, apply AM-GM on $1, 2, 2^2......., 2^n-1$. $$frac1 + 2 + 2^2 +......+2^n-1n geqslant sqrt[n][1*2*....*2^n-1]$$
$$implies frac2^n - 1n geqslant sqrt[n][2^n*(n-1)/2]$$
$$implies frac2^n - 1n geqslant sqrt[2][2^n-1]$$
$$implies 2^n geqslant 1 + n*sqrt[2][2^n-1]$$
Equality holds at n=1.
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
thank you so much Prakhar Neema
$endgroup$
– Rishi Dev Parmar
Mar 13 at 19:13
add a comment |
$begingroup$
If n $epsilon mathbbN$, apply AM-GM on $1, 2, 2^2......., 2^n-1$. $$frac1 + 2 + 2^2 +......+2^n-1n geqslant sqrt[n][1*2*....*2^n-1]$$
$$implies frac2^n - 1n geqslant sqrt[n][2^n*(n-1)/2]$$
$$implies frac2^n - 1n geqslant sqrt[2][2^n-1]$$
$$implies 2^n geqslant 1 + n*sqrt[2][2^n-1]$$
Equality holds at n=1.
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
If n $epsilon mathbbN$, apply AM-GM on $1, 2, 2^2......., 2^n-1$. $$frac1 + 2 + 2^2 +......+2^n-1n geqslant sqrt[n][1*2*....*2^n-1]$$
$$implies frac2^n - 1n geqslant sqrt[n][2^n*(n-1)/2]$$
$$implies frac2^n - 1n geqslant sqrt[2][2^n-1]$$
$$implies 2^n geqslant 1 + n*sqrt[2][2^n-1]$$
Equality holds at n=1.
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
answered Mar 13 at 19:11
Prakhar NeemaPrakhar Neema
1074
1074
New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Prakhar Neema is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
thank you so much Prakhar Neema
$endgroup$
– Rishi Dev Parmar
Mar 13 at 19:13
add a comment |
$begingroup$
thank you so much Prakhar Neema
$endgroup$
– Rishi Dev Parmar
Mar 13 at 19:13
$begingroup$
thank you so much Prakhar Neema
$endgroup$
– Rishi Dev Parmar
Mar 13 at 19:13
$begingroup$
thank you so much Prakhar Neema
$endgroup$
– Rishi Dev Parmar
Mar 13 at 19:13
add a comment |
Rishi Dev Parmar is a new contributor. Be nice, and check out our Code of Conduct.
Rishi Dev Parmar is a new contributor. Be nice, and check out our Code of Conduct.
Rishi Dev Parmar is a new contributor. Be nice, and check out our Code of Conduct.
Rishi Dev Parmar is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
Does this look like a tight inequality for large $n$?
$endgroup$
– Mark Bennet
Mar 13 at 18:37
$begingroup$
Welcome to Mathematics Stack Exchange! Your post is going to attract downvotes with the upper case sentences. A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
Mar 13 at 18:41
1
$begingroup$
Write $2^n-1$ as a sum of powers of two.
$endgroup$
– Mindlack
Mar 13 at 18:47
$begingroup$
Is n an integer?
$endgroup$
– Prakhar Neema
Mar 13 at 18:51
$begingroup$
i don't know what n is as it is not given in the question i think n is a real number.
$endgroup$
– Rishi Dev Parmar
Mar 13 at 18:56