Prove that if $pin{mathbb{R}},;;p>1$ then $(1+x)^p > 1+x^p$ with $x>0$
$begingroup$
I have to prove that if $pin{mathbb{R}},;;p>1$, then $(1+x)^p > 1+x^p$ with $x>0$.
I have got it using derivatives but I am looking for a smarter proof.
Can anyone help me?
Thanks.
real-analysis analysis
$endgroup$
add a comment |
$begingroup$
I have to prove that if $pin{mathbb{R}},;;p>1$, then $(1+x)^p > 1+x^p$ with $x>0$.
I have got it using derivatives but I am looking for a smarter proof.
Can anyone help me?
Thanks.
real-analysis analysis
$endgroup$
4
$begingroup$
Could you share your proof using derivatives? It is hard to know what you mean by "smarter" without seeing it
$endgroup$
– pwerth
Dec 24 '18 at 17:24
1
$begingroup$
I mean that I looking for a prove without using derivatives.
$endgroup$
– mathlife
Dec 24 '18 at 17:27
1
$begingroup$
See the proof using geometric series for the generalized form of Bernulli's inequality.
$endgroup$
– zipirovich
Dec 24 '18 at 18:01
add a comment |
$begingroup$
I have to prove that if $pin{mathbb{R}},;;p>1$, then $(1+x)^p > 1+x^p$ with $x>0$.
I have got it using derivatives but I am looking for a smarter proof.
Can anyone help me?
Thanks.
real-analysis analysis
$endgroup$
I have to prove that if $pin{mathbb{R}},;;p>1$, then $(1+x)^p > 1+x^p$ with $x>0$.
I have got it using derivatives but I am looking for a smarter proof.
Can anyone help me?
Thanks.
real-analysis analysis
real-analysis analysis
edited Dec 31 '18 at 16:40
amWhy
1
1
asked Dec 24 '18 at 17:23
mathlifemathlife
629
629
4
$begingroup$
Could you share your proof using derivatives? It is hard to know what you mean by "smarter" without seeing it
$endgroup$
– pwerth
Dec 24 '18 at 17:24
1
$begingroup$
I mean that I looking for a prove without using derivatives.
$endgroup$
– mathlife
Dec 24 '18 at 17:27
1
$begingroup$
See the proof using geometric series for the generalized form of Bernulli's inequality.
$endgroup$
– zipirovich
Dec 24 '18 at 18:01
add a comment |
4
$begingroup$
Could you share your proof using derivatives? It is hard to know what you mean by "smarter" without seeing it
$endgroup$
– pwerth
Dec 24 '18 at 17:24
1
$begingroup$
I mean that I looking for a prove without using derivatives.
$endgroup$
– mathlife
Dec 24 '18 at 17:27
1
$begingroup$
See the proof using geometric series for the generalized form of Bernulli's inequality.
$endgroup$
– zipirovich
Dec 24 '18 at 18:01
4
4
$begingroup$
Could you share your proof using derivatives? It is hard to know what you mean by "smarter" without seeing it
$endgroup$
– pwerth
Dec 24 '18 at 17:24
$begingroup$
Could you share your proof using derivatives? It is hard to know what you mean by "smarter" without seeing it
$endgroup$
– pwerth
Dec 24 '18 at 17:24
1
1
$begingroup$
I mean that I looking for a prove without using derivatives.
$endgroup$
– mathlife
Dec 24 '18 at 17:27
$begingroup$
I mean that I looking for a prove without using derivatives.
$endgroup$
– mathlife
Dec 24 '18 at 17:27
1
1
$begingroup$
See the proof using geometric series for the generalized form of Bernulli's inequality.
$endgroup$
– zipirovich
Dec 24 '18 at 18:01
$begingroup$
See the proof using geometric series for the generalized form of Bernulli's inequality.
$endgroup$
– zipirovich
Dec 24 '18 at 18:01
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Without loss of generality, we can take $0<xleq1$. Then
$$
(1+x)^pgeq1+pxgeq1+px^p>1+x^p ,
$$
where we used a generalized Bernoulli's inequality in the first step,
the simple comparison $x^pleq x$ in the second step,
and $p>1$ in the last step.
Update: The linked Wiki page does not seem to have an explicit proof of the generalized Bernoulli inequality for real $p>1$, so let me include a proof here that uses properties of logarithm. We have
$$
frac1plog(1+x)
=frac{p-1}plog1+frac1plog(1+x)
leqlogbig(frac{p-1}p+frac{1+x}pbig)
=logbig(1+frac{x}pbig) ,
$$
where we have used the concavity of log in the second step. Now the monotonicity of log gives
$$
(1+x)^{frac1p}leq1+frac{x}p ,
$$
and the substitution $y=frac{x}p$ finishes the job.
General comment: Since this question involves real exponent $p$, and $x^p$ can be defined only for rational $p$ by using a finite process, the proof must involve an infinite process in some way. Thus I would argue that in this situation, using a derivative is not more "shameful" than any other proof. In the particular proof above, the infinite process is hidden under the generalized Bernoulli inequality.
$endgroup$
add a comment |
$begingroup$
Hint:
Use Bernoulli's inequality:
- If $0<xle1$, then $;(1+x)^p>1+px >1+(x^{p-1}),x$ since $p>1ge x^{p-1}$.
- If $x>1$, set $;x=smash{dfrac1u}$ ($0<u<1$). It amounts to checking $;(1+u)^{p}>1+u^p$, which is the previous case.
$endgroup$
$begingroup$
That only works for integer $p$, while the OP says that $pinmathbb{R}$.
$endgroup$
– zipirovich
Dec 24 '18 at 17:54
$begingroup$
Oops! I missed that detail!
$endgroup$
– Bernard
Dec 24 '18 at 18:01
$begingroup$
upvoted to balance the existing downvote
$endgroup$
– timur
Dec 24 '18 at 18:02
2
$begingroup$
Thanks, @Timur! Anyway, I modified the answer. Happy Xmas for the maniac downvoter!
$endgroup$
– Bernard
Dec 24 '18 at 18:11
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Without loss of generality, we can take $0<xleq1$. Then
$$
(1+x)^pgeq1+pxgeq1+px^p>1+x^p ,
$$
where we used a generalized Bernoulli's inequality in the first step,
the simple comparison $x^pleq x$ in the second step,
and $p>1$ in the last step.
Update: The linked Wiki page does not seem to have an explicit proof of the generalized Bernoulli inequality for real $p>1$, so let me include a proof here that uses properties of logarithm. We have
$$
frac1plog(1+x)
=frac{p-1}plog1+frac1plog(1+x)
leqlogbig(frac{p-1}p+frac{1+x}pbig)
=logbig(1+frac{x}pbig) ,
$$
where we have used the concavity of log in the second step. Now the monotonicity of log gives
$$
(1+x)^{frac1p}leq1+frac{x}p ,
$$
and the substitution $y=frac{x}p$ finishes the job.
General comment: Since this question involves real exponent $p$, and $x^p$ can be defined only for rational $p$ by using a finite process, the proof must involve an infinite process in some way. Thus I would argue that in this situation, using a derivative is not more "shameful" than any other proof. In the particular proof above, the infinite process is hidden under the generalized Bernoulli inequality.
$endgroup$
add a comment |
$begingroup$
Without loss of generality, we can take $0<xleq1$. Then
$$
(1+x)^pgeq1+pxgeq1+px^p>1+x^p ,
$$
where we used a generalized Bernoulli's inequality in the first step,
the simple comparison $x^pleq x$ in the second step,
and $p>1$ in the last step.
Update: The linked Wiki page does not seem to have an explicit proof of the generalized Bernoulli inequality for real $p>1$, so let me include a proof here that uses properties of logarithm. We have
$$
frac1plog(1+x)
=frac{p-1}plog1+frac1plog(1+x)
leqlogbig(frac{p-1}p+frac{1+x}pbig)
=logbig(1+frac{x}pbig) ,
$$
where we have used the concavity of log in the second step. Now the monotonicity of log gives
$$
(1+x)^{frac1p}leq1+frac{x}p ,
$$
and the substitution $y=frac{x}p$ finishes the job.
General comment: Since this question involves real exponent $p$, and $x^p$ can be defined only for rational $p$ by using a finite process, the proof must involve an infinite process in some way. Thus I would argue that in this situation, using a derivative is not more "shameful" than any other proof. In the particular proof above, the infinite process is hidden under the generalized Bernoulli inequality.
$endgroup$
add a comment |
$begingroup$
Without loss of generality, we can take $0<xleq1$. Then
$$
(1+x)^pgeq1+pxgeq1+px^p>1+x^p ,
$$
where we used a generalized Bernoulli's inequality in the first step,
the simple comparison $x^pleq x$ in the second step,
and $p>1$ in the last step.
Update: The linked Wiki page does not seem to have an explicit proof of the generalized Bernoulli inequality for real $p>1$, so let me include a proof here that uses properties of logarithm. We have
$$
frac1plog(1+x)
=frac{p-1}plog1+frac1plog(1+x)
leqlogbig(frac{p-1}p+frac{1+x}pbig)
=logbig(1+frac{x}pbig) ,
$$
where we have used the concavity of log in the second step. Now the monotonicity of log gives
$$
(1+x)^{frac1p}leq1+frac{x}p ,
$$
and the substitution $y=frac{x}p$ finishes the job.
General comment: Since this question involves real exponent $p$, and $x^p$ can be defined only for rational $p$ by using a finite process, the proof must involve an infinite process in some way. Thus I would argue that in this situation, using a derivative is not more "shameful" than any other proof. In the particular proof above, the infinite process is hidden under the generalized Bernoulli inequality.
$endgroup$
Without loss of generality, we can take $0<xleq1$. Then
$$
(1+x)^pgeq1+pxgeq1+px^p>1+x^p ,
$$
where we used a generalized Bernoulli's inequality in the first step,
the simple comparison $x^pleq x$ in the second step,
and $p>1$ in the last step.
Update: The linked Wiki page does not seem to have an explicit proof of the generalized Bernoulli inequality for real $p>1$, so let me include a proof here that uses properties of logarithm. We have
$$
frac1plog(1+x)
=frac{p-1}plog1+frac1plog(1+x)
leqlogbig(frac{p-1}p+frac{1+x}pbig)
=logbig(1+frac{x}pbig) ,
$$
where we have used the concavity of log in the second step. Now the monotonicity of log gives
$$
(1+x)^{frac1p}leq1+frac{x}p ,
$$
and the substitution $y=frac{x}p$ finishes the job.
General comment: Since this question involves real exponent $p$, and $x^p$ can be defined only for rational $p$ by using a finite process, the proof must involve an infinite process in some way. Thus I would argue that in this situation, using a derivative is not more "shameful" than any other proof. In the particular proof above, the infinite process is hidden under the generalized Bernoulli inequality.
edited Dec 24 '18 at 19:26
answered Dec 24 '18 at 18:10
timurtimur
12.1k2144
12.1k2144
add a comment |
add a comment |
$begingroup$
Hint:
Use Bernoulli's inequality:
- If $0<xle1$, then $;(1+x)^p>1+px >1+(x^{p-1}),x$ since $p>1ge x^{p-1}$.
- If $x>1$, set $;x=smash{dfrac1u}$ ($0<u<1$). It amounts to checking $;(1+u)^{p}>1+u^p$, which is the previous case.
$endgroup$
$begingroup$
That only works for integer $p$, while the OP says that $pinmathbb{R}$.
$endgroup$
– zipirovich
Dec 24 '18 at 17:54
$begingroup$
Oops! I missed that detail!
$endgroup$
– Bernard
Dec 24 '18 at 18:01
$begingroup$
upvoted to balance the existing downvote
$endgroup$
– timur
Dec 24 '18 at 18:02
2
$begingroup$
Thanks, @Timur! Anyway, I modified the answer. Happy Xmas for the maniac downvoter!
$endgroup$
– Bernard
Dec 24 '18 at 18:11
add a comment |
$begingroup$
Hint:
Use Bernoulli's inequality:
- If $0<xle1$, then $;(1+x)^p>1+px >1+(x^{p-1}),x$ since $p>1ge x^{p-1}$.
- If $x>1$, set $;x=smash{dfrac1u}$ ($0<u<1$). It amounts to checking $;(1+u)^{p}>1+u^p$, which is the previous case.
$endgroup$
$begingroup$
That only works for integer $p$, while the OP says that $pinmathbb{R}$.
$endgroup$
– zipirovich
Dec 24 '18 at 17:54
$begingroup$
Oops! I missed that detail!
$endgroup$
– Bernard
Dec 24 '18 at 18:01
$begingroup$
upvoted to balance the existing downvote
$endgroup$
– timur
Dec 24 '18 at 18:02
2
$begingroup$
Thanks, @Timur! Anyway, I modified the answer. Happy Xmas for the maniac downvoter!
$endgroup$
– Bernard
Dec 24 '18 at 18:11
add a comment |
$begingroup$
Hint:
Use Bernoulli's inequality:
- If $0<xle1$, then $;(1+x)^p>1+px >1+(x^{p-1}),x$ since $p>1ge x^{p-1}$.
- If $x>1$, set $;x=smash{dfrac1u}$ ($0<u<1$). It amounts to checking $;(1+u)^{p}>1+u^p$, which is the previous case.
$endgroup$
Hint:
Use Bernoulli's inequality:
- If $0<xle1$, then $;(1+x)^p>1+px >1+(x^{p-1}),x$ since $p>1ge x^{p-1}$.
- If $x>1$, set $;x=smash{dfrac1u}$ ($0<u<1$). It amounts to checking $;(1+u)^{p}>1+u^p$, which is the previous case.
edited Dec 24 '18 at 18:09
answered Dec 24 '18 at 17:41
BernardBernard
121k740116
121k740116
$begingroup$
That only works for integer $p$, while the OP says that $pinmathbb{R}$.
$endgroup$
– zipirovich
Dec 24 '18 at 17:54
$begingroup$
Oops! I missed that detail!
$endgroup$
– Bernard
Dec 24 '18 at 18:01
$begingroup$
upvoted to balance the existing downvote
$endgroup$
– timur
Dec 24 '18 at 18:02
2
$begingroup$
Thanks, @Timur! Anyway, I modified the answer. Happy Xmas for the maniac downvoter!
$endgroup$
– Bernard
Dec 24 '18 at 18:11
add a comment |
$begingroup$
That only works for integer $p$, while the OP says that $pinmathbb{R}$.
$endgroup$
– zipirovich
Dec 24 '18 at 17:54
$begingroup$
Oops! I missed that detail!
$endgroup$
– Bernard
Dec 24 '18 at 18:01
$begingroup$
upvoted to balance the existing downvote
$endgroup$
– timur
Dec 24 '18 at 18:02
2
$begingroup$
Thanks, @Timur! Anyway, I modified the answer. Happy Xmas for the maniac downvoter!
$endgroup$
– Bernard
Dec 24 '18 at 18:11
$begingroup$
That only works for integer $p$, while the OP says that $pinmathbb{R}$.
$endgroup$
– zipirovich
Dec 24 '18 at 17:54
$begingroup$
That only works for integer $p$, while the OP says that $pinmathbb{R}$.
$endgroup$
– zipirovich
Dec 24 '18 at 17:54
$begingroup$
Oops! I missed that detail!
$endgroup$
– Bernard
Dec 24 '18 at 18:01
$begingroup$
Oops! I missed that detail!
$endgroup$
– Bernard
Dec 24 '18 at 18:01
$begingroup$
upvoted to balance the existing downvote
$endgroup$
– timur
Dec 24 '18 at 18:02
$begingroup$
upvoted to balance the existing downvote
$endgroup$
– timur
Dec 24 '18 at 18:02
2
2
$begingroup$
Thanks, @Timur! Anyway, I modified the answer. Happy Xmas for the maniac downvoter!
$endgroup$
– Bernard
Dec 24 '18 at 18:11
$begingroup$
Thanks, @Timur! Anyway, I modified the answer. Happy Xmas for the maniac downvoter!
$endgroup$
– Bernard
Dec 24 '18 at 18:11
add a comment |
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4
$begingroup$
Could you share your proof using derivatives? It is hard to know what you mean by "smarter" without seeing it
$endgroup$
– pwerth
Dec 24 '18 at 17:24
1
$begingroup$
I mean that I looking for a prove without using derivatives.
$endgroup$
– mathlife
Dec 24 '18 at 17:27
1
$begingroup$
See the proof using geometric series for the generalized form of Bernulli's inequality.
$endgroup$
– zipirovich
Dec 24 '18 at 18:01