Show that $lim_{ntoinfty}frac{ln(n!)}{n} = +infty$
$begingroup$
Show that
$$
lim_{ntoinfty}frac{ln(n!)}{n} = +infty
$$
The only way i've been able to show that is using Stirling's approximation:
$$
n! simsqrt{2pi n}left(frac{n}{e}right)^n
$$
Let:
$$
begin{cases}
x_n = frac{ln(n!)}{n}\
n in Bbb N
end{cases}
$$
So we may rewrite $x_n$ as:
$$
x_n sim frac{ln(2pi n)}{2n} + frac{nln(frac{n}{e})}{n}
$$
Now using the fact that $lim(x_n + y_n) = lim x_n + lim y_n$ :
$$
lim_{ntoinfty}x_n = lim_{ntoinfty}frac{ln(2pi n)}{2n} + lim_{ntoinfty}frac{nln(frac{n}{e})}{n} = 0 + infty
$$
I'm looking for another way to show this, since Stirling's approximation has not been introduced at the point where i took the exercise from yet.
calculus sequences-and-series limits factorial
$endgroup$
add a comment |
$begingroup$
Show that
$$
lim_{ntoinfty}frac{ln(n!)}{n} = +infty
$$
The only way i've been able to show that is using Stirling's approximation:
$$
n! simsqrt{2pi n}left(frac{n}{e}right)^n
$$
Let:
$$
begin{cases}
x_n = frac{ln(n!)}{n}\
n in Bbb N
end{cases}
$$
So we may rewrite $x_n$ as:
$$
x_n sim frac{ln(2pi n)}{2n} + frac{nln(frac{n}{e})}{n}
$$
Now using the fact that $lim(x_n + y_n) = lim x_n + lim y_n$ :
$$
lim_{ntoinfty}x_n = lim_{ntoinfty}frac{ln(2pi n)}{2n} + lim_{ntoinfty}frac{nln(frac{n}{e})}{n} = 0 + infty
$$
I'm looking for another way to show this, since Stirling's approximation has not been introduced at the point where i took the exercise from yet.
calculus sequences-and-series limits factorial
$endgroup$
$begingroup$
Cesaro-Stolz is the key here.
$endgroup$
– Paramanand Singh
Dec 8 '18 at 7:34
add a comment |
$begingroup$
Show that
$$
lim_{ntoinfty}frac{ln(n!)}{n} = +infty
$$
The only way i've been able to show that is using Stirling's approximation:
$$
n! simsqrt{2pi n}left(frac{n}{e}right)^n
$$
Let:
$$
begin{cases}
x_n = frac{ln(n!)}{n}\
n in Bbb N
end{cases}
$$
So we may rewrite $x_n$ as:
$$
x_n sim frac{ln(2pi n)}{2n} + frac{nln(frac{n}{e})}{n}
$$
Now using the fact that $lim(x_n + y_n) = lim x_n + lim y_n$ :
$$
lim_{ntoinfty}x_n = lim_{ntoinfty}frac{ln(2pi n)}{2n} + lim_{ntoinfty}frac{nln(frac{n}{e})}{n} = 0 + infty
$$
I'm looking for another way to show this, since Stirling's approximation has not been introduced at the point where i took the exercise from yet.
calculus sequences-and-series limits factorial
$endgroup$
Show that
$$
lim_{ntoinfty}frac{ln(n!)}{n} = +infty
$$
The only way i've been able to show that is using Stirling's approximation:
$$
n! simsqrt{2pi n}left(frac{n}{e}right)^n
$$
Let:
$$
begin{cases}
x_n = frac{ln(n!)}{n}\
n in Bbb N
end{cases}
$$
So we may rewrite $x_n$ as:
$$
x_n sim frac{ln(2pi n)}{2n} + frac{nln(frac{n}{e})}{n}
$$
Now using the fact that $lim(x_n + y_n) = lim x_n + lim y_n$ :
$$
lim_{ntoinfty}x_n = lim_{ntoinfty}frac{ln(2pi n)}{2n} + lim_{ntoinfty}frac{nln(frac{n}{e})}{n} = 0 + infty
$$
I'm looking for another way to show this, since Stirling's approximation has not been introduced at the point where i took the exercise from yet.
calculus sequences-and-series limits factorial
calculus sequences-and-series limits factorial
asked Dec 7 '18 at 15:50
romanroman
2,14421222
2,14421222
$begingroup$
Cesaro-Stolz is the key here.
$endgroup$
– Paramanand Singh
Dec 8 '18 at 7:34
add a comment |
$begingroup$
Cesaro-Stolz is the key here.
$endgroup$
– Paramanand Singh
Dec 8 '18 at 7:34
$begingroup$
Cesaro-Stolz is the key here.
$endgroup$
– Paramanand Singh
Dec 8 '18 at 7:34
$begingroup$
Cesaro-Stolz is the key here.
$endgroup$
– Paramanand Singh
Dec 8 '18 at 7:34
add a comment |
5 Answers
5
active
oldest
votes
$begingroup$
Another way to show $$lim_{ntoinfty}frac{ln(n!)}{n}=infty$$ is to consider the following property of logarithms: $$log(n!)=log(n)+log(n-1)+cdots+log(2)>frac{n}{2}logleft(frac n2right).$$ Now $$frac{log (n!)}{n}>frac{log(n/2)}{2}.$$ As $ntoinfty$, this clearly diverges to $+infty$.
$endgroup$
1
$begingroup$
Thank you for the answer, could you please elaborate on $ln(n!) > frac{n}{2}lnleft(frac{n}{2}right)$?
$endgroup$
– roman
Dec 7 '18 at 15:59
3
$begingroup$
@roman: There are $n-1$ terms in the sum after expanding the logarithm (therefore, there are at least $n/2$ terms for $n>3$...this is where the first $n/2$ comes from). Now, if we take the first $n/2$ terms in the sum, the argument of the logarithms in all of these terms is at least $n/2$ (that is, $log(n)>log(n/2)$, $log(n-1)>log(n/2)$, etc.)
$endgroup$
– Clayton
Dec 7 '18 at 16:05
add a comment |
$begingroup$
Hint: $$ln(n!)=ln 1+ln2+cdots+ln n$$For all but the smallest $n$, most of those terms are larger than $1$, so the sum is larger than $n$.
For somewhat large $n$, most of those terms are larger than $2$, so the sum is larger than $2n$.
For even larger $n$, most of those terms are larger than $3$, so the sum is larger than $3n$.
$endgroup$
add a comment |
$begingroup$
The Cesaro-Stolz criterion is your easiest and cleanest way out of this. It states that given sequences $x, y in mathbb{R}^{mathbb{N}}$ such that $y$ is strictly increasing and unbounded and the sequence of successive increments converges in the extended real line
$$lim_{n to infty} frac{x_{n+1}-x_{n}}{y_{n+1}-y_{n}}=t in overline{mathbb{R}}$$
then $$lim_{n to infty}frac{x_{n}}{y_{n}}=t$$
You can apply this to $x=(mathrm{ln}(n!))_{n in mathbb{N}}$ and $y=(n)_{n in mathbb{N}}$.
A similar argument would rely on a version of the ratio criterion: if $x in (0, infty)^{mathbb{N}}$ is a sequence of strictly positive reals such that
$$lim_{n to infty} frac{x_{n+1}}{x_n}=a in [0, infty]$$
then
$$lim_{n to infty} sqrt [n]{x_{n}}=a$$
This criterion itself can be proved by the Cesaro-Stolz criterion (there are also other methods) and you can apply it to conclude that
$$sqrt[n]{n!} xrightarrow{n to infty} infty$$
as $frac{(n+1)!}{n!}=n+1 xrightarrow{n to infty} infty$.
In the same vein of employing ratios, one can settle the convergence of the sequence
$$left(frac{sqrt[n]{n!}}{n}right)_{n in mathbb{N}^{*}}=left(sqrt[n]{frac{n!}{n^n}}right)_{n in mathbb{N}^{*}}$$
by studying the sequence of successive ratios:
$$ frac{(n+1)!}{(n+1)^{n+1}} cdot frac{n^n}{n!}=left(frac{n}{n+1}right)^{n} xrightarrow{n to infty} frac{1}{mathrm{e}}$$
Hence,
$$sqrt[n]{n!}=n cdot frac{sqrt[n]{n!}}{n} xrightarrow{n to infty} infty cdot frac{1}{mathrm{e}}=infty$$
$endgroup$
add a comment |
$begingroup$
We have that
$$ln(n!)ge n ln n - n$$
then
$$frac{ln(n!)}{n}ge ln n -1 to infty$$
For the proof of the first inequality refer to
- Prove that $n ln(n) - n le ln(n!)$ without Stirling
$endgroup$
add a comment |
$begingroup$
Here is another way considering
- $e^{frac{ln n!}{n}}$
begin{eqnarray*} e^{frac{ln n!}{n}}
& = & sqrt[n]{n!}\
& stackrel{GM-HM}{geq} & frac{n}{frac{1}{1}+cdots frac{1}{n}} \
& stackrel{sum_{k=1}^n frac{1}{k} < ln n + 1}{>} & frac{n}{ln n +1} \
& stackrel{n to infty}{longrightarrow} & +infty
end{eqnarray*}
$endgroup$
add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Another way to show $$lim_{ntoinfty}frac{ln(n!)}{n}=infty$$ is to consider the following property of logarithms: $$log(n!)=log(n)+log(n-1)+cdots+log(2)>frac{n}{2}logleft(frac n2right).$$ Now $$frac{log (n!)}{n}>frac{log(n/2)}{2}.$$ As $ntoinfty$, this clearly diverges to $+infty$.
$endgroup$
1
$begingroup$
Thank you for the answer, could you please elaborate on $ln(n!) > frac{n}{2}lnleft(frac{n}{2}right)$?
$endgroup$
– roman
Dec 7 '18 at 15:59
3
$begingroup$
@roman: There are $n-1$ terms in the sum after expanding the logarithm (therefore, there are at least $n/2$ terms for $n>3$...this is where the first $n/2$ comes from). Now, if we take the first $n/2$ terms in the sum, the argument of the logarithms in all of these terms is at least $n/2$ (that is, $log(n)>log(n/2)$, $log(n-1)>log(n/2)$, etc.)
$endgroup$
– Clayton
Dec 7 '18 at 16:05
add a comment |
$begingroup$
Another way to show $$lim_{ntoinfty}frac{ln(n!)}{n}=infty$$ is to consider the following property of logarithms: $$log(n!)=log(n)+log(n-1)+cdots+log(2)>frac{n}{2}logleft(frac n2right).$$ Now $$frac{log (n!)}{n}>frac{log(n/2)}{2}.$$ As $ntoinfty$, this clearly diverges to $+infty$.
$endgroup$
1
$begingroup$
Thank you for the answer, could you please elaborate on $ln(n!) > frac{n}{2}lnleft(frac{n}{2}right)$?
$endgroup$
– roman
Dec 7 '18 at 15:59
3
$begingroup$
@roman: There are $n-1$ terms in the sum after expanding the logarithm (therefore, there are at least $n/2$ terms for $n>3$...this is where the first $n/2$ comes from). Now, if we take the first $n/2$ terms in the sum, the argument of the logarithms in all of these terms is at least $n/2$ (that is, $log(n)>log(n/2)$, $log(n-1)>log(n/2)$, etc.)
$endgroup$
– Clayton
Dec 7 '18 at 16:05
add a comment |
$begingroup$
Another way to show $$lim_{ntoinfty}frac{ln(n!)}{n}=infty$$ is to consider the following property of logarithms: $$log(n!)=log(n)+log(n-1)+cdots+log(2)>frac{n}{2}logleft(frac n2right).$$ Now $$frac{log (n!)}{n}>frac{log(n/2)}{2}.$$ As $ntoinfty$, this clearly diverges to $+infty$.
$endgroup$
Another way to show $$lim_{ntoinfty}frac{ln(n!)}{n}=infty$$ is to consider the following property of logarithms: $$log(n!)=log(n)+log(n-1)+cdots+log(2)>frac{n}{2}logleft(frac n2right).$$ Now $$frac{log (n!)}{n}>frac{log(n/2)}{2}.$$ As $ntoinfty$, this clearly diverges to $+infty$.
answered Dec 7 '18 at 15:54
ClaytonClayton
19.1k33185
19.1k33185
1
$begingroup$
Thank you for the answer, could you please elaborate on $ln(n!) > frac{n}{2}lnleft(frac{n}{2}right)$?
$endgroup$
– roman
Dec 7 '18 at 15:59
3
$begingroup$
@roman: There are $n-1$ terms in the sum after expanding the logarithm (therefore, there are at least $n/2$ terms for $n>3$...this is where the first $n/2$ comes from). Now, if we take the first $n/2$ terms in the sum, the argument of the logarithms in all of these terms is at least $n/2$ (that is, $log(n)>log(n/2)$, $log(n-1)>log(n/2)$, etc.)
$endgroup$
– Clayton
Dec 7 '18 at 16:05
add a comment |
1
$begingroup$
Thank you for the answer, could you please elaborate on $ln(n!) > frac{n}{2}lnleft(frac{n}{2}right)$?
$endgroup$
– roman
Dec 7 '18 at 15:59
3
$begingroup$
@roman: There are $n-1$ terms in the sum after expanding the logarithm (therefore, there are at least $n/2$ terms for $n>3$...this is where the first $n/2$ comes from). Now, if we take the first $n/2$ terms in the sum, the argument of the logarithms in all of these terms is at least $n/2$ (that is, $log(n)>log(n/2)$, $log(n-1)>log(n/2)$, etc.)
$endgroup$
– Clayton
Dec 7 '18 at 16:05
1
1
$begingroup$
Thank you for the answer, could you please elaborate on $ln(n!) > frac{n}{2}lnleft(frac{n}{2}right)$?
$endgroup$
– roman
Dec 7 '18 at 15:59
$begingroup$
Thank you for the answer, could you please elaborate on $ln(n!) > frac{n}{2}lnleft(frac{n}{2}right)$?
$endgroup$
– roman
Dec 7 '18 at 15:59
3
3
$begingroup$
@roman: There are $n-1$ terms in the sum after expanding the logarithm (therefore, there are at least $n/2$ terms for $n>3$...this is where the first $n/2$ comes from). Now, if we take the first $n/2$ terms in the sum, the argument of the logarithms in all of these terms is at least $n/2$ (that is, $log(n)>log(n/2)$, $log(n-1)>log(n/2)$, etc.)
$endgroup$
– Clayton
Dec 7 '18 at 16:05
$begingroup$
@roman: There are $n-1$ terms in the sum after expanding the logarithm (therefore, there are at least $n/2$ terms for $n>3$...this is where the first $n/2$ comes from). Now, if we take the first $n/2$ terms in the sum, the argument of the logarithms in all of these terms is at least $n/2$ (that is, $log(n)>log(n/2)$, $log(n-1)>log(n/2)$, etc.)
$endgroup$
– Clayton
Dec 7 '18 at 16:05
add a comment |
$begingroup$
Hint: $$ln(n!)=ln 1+ln2+cdots+ln n$$For all but the smallest $n$, most of those terms are larger than $1$, so the sum is larger than $n$.
For somewhat large $n$, most of those terms are larger than $2$, so the sum is larger than $2n$.
For even larger $n$, most of those terms are larger than $3$, so the sum is larger than $3n$.
$endgroup$
add a comment |
$begingroup$
Hint: $$ln(n!)=ln 1+ln2+cdots+ln n$$For all but the smallest $n$, most of those terms are larger than $1$, so the sum is larger than $n$.
For somewhat large $n$, most of those terms are larger than $2$, so the sum is larger than $2n$.
For even larger $n$, most of those terms are larger than $3$, so the sum is larger than $3n$.
$endgroup$
add a comment |
$begingroup$
Hint: $$ln(n!)=ln 1+ln2+cdots+ln n$$For all but the smallest $n$, most of those terms are larger than $1$, so the sum is larger than $n$.
For somewhat large $n$, most of those terms are larger than $2$, so the sum is larger than $2n$.
For even larger $n$, most of those terms are larger than $3$, so the sum is larger than $3n$.
$endgroup$
Hint: $$ln(n!)=ln 1+ln2+cdots+ln n$$For all but the smallest $n$, most of those terms are larger than $1$, so the sum is larger than $n$.
For somewhat large $n$, most of those terms are larger than $2$, so the sum is larger than $2n$.
For even larger $n$, most of those terms are larger than $3$, so the sum is larger than $3n$.
answered Dec 7 '18 at 15:54
ArthurArthur
113k7113196
113k7113196
add a comment |
add a comment |
$begingroup$
The Cesaro-Stolz criterion is your easiest and cleanest way out of this. It states that given sequences $x, y in mathbb{R}^{mathbb{N}}$ such that $y$ is strictly increasing and unbounded and the sequence of successive increments converges in the extended real line
$$lim_{n to infty} frac{x_{n+1}-x_{n}}{y_{n+1}-y_{n}}=t in overline{mathbb{R}}$$
then $$lim_{n to infty}frac{x_{n}}{y_{n}}=t$$
You can apply this to $x=(mathrm{ln}(n!))_{n in mathbb{N}}$ and $y=(n)_{n in mathbb{N}}$.
A similar argument would rely on a version of the ratio criterion: if $x in (0, infty)^{mathbb{N}}$ is a sequence of strictly positive reals such that
$$lim_{n to infty} frac{x_{n+1}}{x_n}=a in [0, infty]$$
then
$$lim_{n to infty} sqrt [n]{x_{n}}=a$$
This criterion itself can be proved by the Cesaro-Stolz criterion (there are also other methods) and you can apply it to conclude that
$$sqrt[n]{n!} xrightarrow{n to infty} infty$$
as $frac{(n+1)!}{n!}=n+1 xrightarrow{n to infty} infty$.
In the same vein of employing ratios, one can settle the convergence of the sequence
$$left(frac{sqrt[n]{n!}}{n}right)_{n in mathbb{N}^{*}}=left(sqrt[n]{frac{n!}{n^n}}right)_{n in mathbb{N}^{*}}$$
by studying the sequence of successive ratios:
$$ frac{(n+1)!}{(n+1)^{n+1}} cdot frac{n^n}{n!}=left(frac{n}{n+1}right)^{n} xrightarrow{n to infty} frac{1}{mathrm{e}}$$
Hence,
$$sqrt[n]{n!}=n cdot frac{sqrt[n]{n!}}{n} xrightarrow{n to infty} infty cdot frac{1}{mathrm{e}}=infty$$
$endgroup$
add a comment |
$begingroup$
The Cesaro-Stolz criterion is your easiest and cleanest way out of this. It states that given sequences $x, y in mathbb{R}^{mathbb{N}}$ such that $y$ is strictly increasing and unbounded and the sequence of successive increments converges in the extended real line
$$lim_{n to infty} frac{x_{n+1}-x_{n}}{y_{n+1}-y_{n}}=t in overline{mathbb{R}}$$
then $$lim_{n to infty}frac{x_{n}}{y_{n}}=t$$
You can apply this to $x=(mathrm{ln}(n!))_{n in mathbb{N}}$ and $y=(n)_{n in mathbb{N}}$.
A similar argument would rely on a version of the ratio criterion: if $x in (0, infty)^{mathbb{N}}$ is a sequence of strictly positive reals such that
$$lim_{n to infty} frac{x_{n+1}}{x_n}=a in [0, infty]$$
then
$$lim_{n to infty} sqrt [n]{x_{n}}=a$$
This criterion itself can be proved by the Cesaro-Stolz criterion (there are also other methods) and you can apply it to conclude that
$$sqrt[n]{n!} xrightarrow{n to infty} infty$$
as $frac{(n+1)!}{n!}=n+1 xrightarrow{n to infty} infty$.
In the same vein of employing ratios, one can settle the convergence of the sequence
$$left(frac{sqrt[n]{n!}}{n}right)_{n in mathbb{N}^{*}}=left(sqrt[n]{frac{n!}{n^n}}right)_{n in mathbb{N}^{*}}$$
by studying the sequence of successive ratios:
$$ frac{(n+1)!}{(n+1)^{n+1}} cdot frac{n^n}{n!}=left(frac{n}{n+1}right)^{n} xrightarrow{n to infty} frac{1}{mathrm{e}}$$
Hence,
$$sqrt[n]{n!}=n cdot frac{sqrt[n]{n!}}{n} xrightarrow{n to infty} infty cdot frac{1}{mathrm{e}}=infty$$
$endgroup$
add a comment |
$begingroup$
The Cesaro-Stolz criterion is your easiest and cleanest way out of this. It states that given sequences $x, y in mathbb{R}^{mathbb{N}}$ such that $y$ is strictly increasing and unbounded and the sequence of successive increments converges in the extended real line
$$lim_{n to infty} frac{x_{n+1}-x_{n}}{y_{n+1}-y_{n}}=t in overline{mathbb{R}}$$
then $$lim_{n to infty}frac{x_{n}}{y_{n}}=t$$
You can apply this to $x=(mathrm{ln}(n!))_{n in mathbb{N}}$ and $y=(n)_{n in mathbb{N}}$.
A similar argument would rely on a version of the ratio criterion: if $x in (0, infty)^{mathbb{N}}$ is a sequence of strictly positive reals such that
$$lim_{n to infty} frac{x_{n+1}}{x_n}=a in [0, infty]$$
then
$$lim_{n to infty} sqrt [n]{x_{n}}=a$$
This criterion itself can be proved by the Cesaro-Stolz criterion (there are also other methods) and you can apply it to conclude that
$$sqrt[n]{n!} xrightarrow{n to infty} infty$$
as $frac{(n+1)!}{n!}=n+1 xrightarrow{n to infty} infty$.
In the same vein of employing ratios, one can settle the convergence of the sequence
$$left(frac{sqrt[n]{n!}}{n}right)_{n in mathbb{N}^{*}}=left(sqrt[n]{frac{n!}{n^n}}right)_{n in mathbb{N}^{*}}$$
by studying the sequence of successive ratios:
$$ frac{(n+1)!}{(n+1)^{n+1}} cdot frac{n^n}{n!}=left(frac{n}{n+1}right)^{n} xrightarrow{n to infty} frac{1}{mathrm{e}}$$
Hence,
$$sqrt[n]{n!}=n cdot frac{sqrt[n]{n!}}{n} xrightarrow{n to infty} infty cdot frac{1}{mathrm{e}}=infty$$
$endgroup$
The Cesaro-Stolz criterion is your easiest and cleanest way out of this. It states that given sequences $x, y in mathbb{R}^{mathbb{N}}$ such that $y$ is strictly increasing and unbounded and the sequence of successive increments converges in the extended real line
$$lim_{n to infty} frac{x_{n+1}-x_{n}}{y_{n+1}-y_{n}}=t in overline{mathbb{R}}$$
then $$lim_{n to infty}frac{x_{n}}{y_{n}}=t$$
You can apply this to $x=(mathrm{ln}(n!))_{n in mathbb{N}}$ and $y=(n)_{n in mathbb{N}}$.
A similar argument would rely on a version of the ratio criterion: if $x in (0, infty)^{mathbb{N}}$ is a sequence of strictly positive reals such that
$$lim_{n to infty} frac{x_{n+1}}{x_n}=a in [0, infty]$$
then
$$lim_{n to infty} sqrt [n]{x_{n}}=a$$
This criterion itself can be proved by the Cesaro-Stolz criterion (there are also other methods) and you can apply it to conclude that
$$sqrt[n]{n!} xrightarrow{n to infty} infty$$
as $frac{(n+1)!}{n!}=n+1 xrightarrow{n to infty} infty$.
In the same vein of employing ratios, one can settle the convergence of the sequence
$$left(frac{sqrt[n]{n!}}{n}right)_{n in mathbb{N}^{*}}=left(sqrt[n]{frac{n!}{n^n}}right)_{n in mathbb{N}^{*}}$$
by studying the sequence of successive ratios:
$$ frac{(n+1)!}{(n+1)^{n+1}} cdot frac{n^n}{n!}=left(frac{n}{n+1}right)^{n} xrightarrow{n to infty} frac{1}{mathrm{e}}$$
Hence,
$$sqrt[n]{n!}=n cdot frac{sqrt[n]{n!}}{n} xrightarrow{n to infty} infty cdot frac{1}{mathrm{e}}=infty$$
edited Dec 8 '18 at 4:58
answered Dec 7 '18 at 16:01
ΑΘΩΑΘΩ
2463
2463
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$begingroup$
We have that
$$ln(n!)ge n ln n - n$$
then
$$frac{ln(n!)}{n}ge ln n -1 to infty$$
For the proof of the first inequality refer to
- Prove that $n ln(n) - n le ln(n!)$ without Stirling
$endgroup$
add a comment |
$begingroup$
We have that
$$ln(n!)ge n ln n - n$$
then
$$frac{ln(n!)}{n}ge ln n -1 to infty$$
For the proof of the first inequality refer to
- Prove that $n ln(n) - n le ln(n!)$ without Stirling
$endgroup$
add a comment |
$begingroup$
We have that
$$ln(n!)ge n ln n - n$$
then
$$frac{ln(n!)}{n}ge ln n -1 to infty$$
For the proof of the first inequality refer to
- Prove that $n ln(n) - n le ln(n!)$ without Stirling
$endgroup$
We have that
$$ln(n!)ge n ln n - n$$
then
$$frac{ln(n!)}{n}ge ln n -1 to infty$$
For the proof of the first inequality refer to
- Prove that $n ln(n) - n le ln(n!)$ without Stirling
answered Dec 7 '18 at 16:23
gimusigimusi
92.8k84494
92.8k84494
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$begingroup$
Here is another way considering
- $e^{frac{ln n!}{n}}$
begin{eqnarray*} e^{frac{ln n!}{n}}
& = & sqrt[n]{n!}\
& stackrel{GM-HM}{geq} & frac{n}{frac{1}{1}+cdots frac{1}{n}} \
& stackrel{sum_{k=1}^n frac{1}{k} < ln n + 1}{>} & frac{n}{ln n +1} \
& stackrel{n to infty}{longrightarrow} & +infty
end{eqnarray*}
$endgroup$
add a comment |
$begingroup$
Here is another way considering
- $e^{frac{ln n!}{n}}$
begin{eqnarray*} e^{frac{ln n!}{n}}
& = & sqrt[n]{n!}\
& stackrel{GM-HM}{geq} & frac{n}{frac{1}{1}+cdots frac{1}{n}} \
& stackrel{sum_{k=1}^n frac{1}{k} < ln n + 1}{>} & frac{n}{ln n +1} \
& stackrel{n to infty}{longrightarrow} & +infty
end{eqnarray*}
$endgroup$
add a comment |
$begingroup$
Here is another way considering
- $e^{frac{ln n!}{n}}$
begin{eqnarray*} e^{frac{ln n!}{n}}
& = & sqrt[n]{n!}\
& stackrel{GM-HM}{geq} & frac{n}{frac{1}{1}+cdots frac{1}{n}} \
& stackrel{sum_{k=1}^n frac{1}{k} < ln n + 1}{>} & frac{n}{ln n +1} \
& stackrel{n to infty}{longrightarrow} & +infty
end{eqnarray*}
$endgroup$
Here is another way considering
- $e^{frac{ln n!}{n}}$
begin{eqnarray*} e^{frac{ln n!}{n}}
& = & sqrt[n]{n!}\
& stackrel{GM-HM}{geq} & frac{n}{frac{1}{1}+cdots frac{1}{n}} \
& stackrel{sum_{k=1}^n frac{1}{k} < ln n + 1}{>} & frac{n}{ln n +1} \
& stackrel{n to infty}{longrightarrow} & +infty
end{eqnarray*}
answered Dec 7 '18 at 16:30
trancelocationtrancelocation
10.6k1722
10.6k1722
add a comment |
add a comment |
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$begingroup$
Cesaro-Stolz is the key here.
$endgroup$
– Paramanand Singh
Dec 8 '18 at 7:34