Why $ lim_nrightarrow infty fracn!n^k(n-k)! =1 $? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1Limits involving factorials $lim_Ntoinfty fracN!(N-k)!N^k$Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$Proof that $limlimits_h to infty frach!h^k(h-k)!=1 $ for any $ k $Evaluating $lim_nrightarrowinftyleft(1-fracxn^1+aright)^n$Computing $lim_nrightarrowinfty(1-fracxn)^-n$Dominated convergence theorem for complex-valued functions?Limit to Expectation: $ - lim_N rightarrow infty frac1N sum_n=1^N fracpartialpartial theta ln p(x_n|theta)$Evaluation of $lim_mtoinftyBig(F(e^-fraclambdam^2)Big)^m$ given $F(z)=frac1-sqrt1-z^2z$Compute $lim_nrightarrowinftyleft(fracn+1nright)^n^2cdotfrac1e^n.$Find $lim_nrightarrow inftyfrac(2n-1)!!(2n)!!.$Finding $limsup_nrightarrowinfty n^fraclog(n)n$How to prove $lim_n rightarrowinfty e^-nsum_k=0^nfracn^kk! = frac12$?Show: $lim_nrightarrow infty left|int_1^eleft[ln(x)right]^n:dx right|= 0 $
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Why $ lim_nrightarrow infty fracn!n^k(n-k)! =1 $?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1Limits involving factorials $lim_Ntoinfty fracN!(N-k)!N^k$Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$Proof that $limlimits_h to infty frach!h^k(h-k)!=1 $ for any $ k $Evaluating $lim_nrightarrowinftyleft(1-fracxn^1+aright)^n$Computing $lim_nrightarrowinfty(1-fracxn)^-n$Dominated convergence theorem for complex-valued functions?Limit to Expectation: $ - lim_N rightarrow infty frac1N sum_n=1^N fracpartialpartial theta ln p(x_n|theta)$Evaluation of $lim_mtoinftyBig(F(e^-fraclambdam^2)Big)^m$ given $F(z)=frac1-sqrt1-z^2z$Compute $lim_nrightarrowinftyleft(fracn+1nright)^n^2cdotfrac1e^n.$Find $lim_nrightarrow inftyfrac(2n-1)!!(2n)!!}.$Finding $limsup_nrightarrowinfty n^{fraclog(n)n$How to prove $lim_n rightarrowinfty e^-nsum_k=0^nfracn^kk! = frac12$?Show: $lim_nrightarrow infty left|int_1^eleft[ln(x)right]^n:dx right|= 0 $
$begingroup$
I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable.
Consider the binomial distribution:
$$
beginequationbeginaligned
P(X=k) &=binom n k p^k (1-p)^n-k\
&=fracn!k!(n-k)! p^k (1-p)^n-k
endalignedendequation
$$
Substitute $m=np$ , or $p=fracmn$ :
$$
beginequationbeginaligned
P(X=k) &=fracn!k!(n-k)! left(fracmnright)^k left(1-fracmnright)^n-k\
&=fracn!k!(n-k)! fracm^kn^k left(1-fracmnright)^nleft(1-fracmnright)^-k
endalignedendequation
$$
Slightly rearrange
$$
beginequationbeginaligned
&=fracn!n^k(n-k)! left(1-fracmnright)^-kfracm^kk!left(1-fracmnright)^n
endalignedendequation
$$
Note that
$$
beginequationbeginaligned
& lim_nrightarrow infty fracn!n^k(n-k)! =1,quadlim_nrightarrow infty left(1-fracmnright)^-k =1,quad lim_nrightarrow infty left(1-fracmnright)^n =e^-m
endalignedendequation
$$
Thus, we have the final result which is equal to the formula for the Poisson distribution.
$$
=fracm^k e^-mk!
$$
In all these steps, what I don't understand is the following limit:
$$
lim_nrightarrow infty fracn!n^k(n-k)! =1
$$
limits factorial
asked 2 days ago
billyandrbillyandr
237
$endgroup$
add a comment |
$begingroup$
I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable.
Consider the binomial distribution:
$$
beginequationbeginaligned
P(X=k) &=binom n k p^k (1-p)^n-k\
&=fracn!k!(n-k)! p^k (1-p)^n-k
endalignedendequation
$$
Substitute $m=np$ , or $p=fracmn$ :
$$
beginequationbeginaligned
P(X=k) &=fracn!k!(n-k)! left(fracmnright)^k left(1-fracmnright)^n-k\
&=fracn!k!(n-k)! fracm^kn^k left(1-fracmnright)^nleft(1-fracmnright)^-k
endalignedendequation
$$
Slightly rearrange
$$
beginequationbeginaligned
&=fracn!n^k(n-k)! left(1-fracmnright)^-kfracm^kk!left(1-fracmnright)^n
endalignedendequation
$$
Note that
$$
beginequationbeginaligned
& lim_nrightarrow infty fracn!n^k(n-k)! =1,quadlim_nrightarrow infty left(1-fracmnright)^-k =1,quad lim_nrightarrow infty left(1-fracmnright)^n =e^-m
endalignedendequation
$$
Thus, we have the final result which is equal to the formula for the Poisson distribution.
$$
=fracm^k e^-mk!
$$
In all these steps, what I don't understand is the following limit:
$$
lim_nrightarrow infty fracn!n^k(n-k)! =1
$$
limits factorial
asked 2 days ago
billyandrbillyandr
237
$endgroup$
$begingroup$
There are several posts about this: Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1, Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$, Proof that $limlimits_h to infty frach!h^k(h-k)!=1$ for any $k$, Limits involing Factorials $lim_Ntoinfty fracN!(N-k)!N^k$
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
I found the posts in the above comment using Approach0. For some useful tips on searching here see: How to search on this site?
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
Some of the other posts treating the same question painfully lack details and context. Maybe you'd want to put them on hold or close them.
$endgroup$
– billyandr
yesterday
$begingroup$
billyandr: If you actually have a look at those links, you can see that two of those posts are closed (as duplicates) now. Let me also say that the fact that you have added some more context to your question is certainly appreciated. (After all, that's what lead to reopening.)
$endgroup$
– Martin Sleziak
yesterday
add a comment |
$begingroup$
I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable.
Consider the binomial distribution:
$$
beginequationbeginaligned
P(X=k) &=binom n k p^k (1-p)^n-k\
&=fracn!k!(n-k)! p^k (1-p)^n-k
endalignedendequation
$$
Substitute $m=np$ , or $p=fracmn$ :
$$
beginequationbeginaligned
P(X=k) &=fracn!k!(n-k)! left(fracmnright)^k left(1-fracmnright)^n-k\
&=fracn!k!(n-k)! fracm^kn^k left(1-fracmnright)^nleft(1-fracmnright)^-k
endalignedendequation
$$
Slightly rearrange
$$
beginequationbeginaligned
&=fracn!n^k(n-k)! left(1-fracmnright)^-kfracm^kk!left(1-fracmnright)^n
endalignedendequation
$$
Note that
$$
beginequationbeginaligned
& lim_nrightarrow infty fracn!n^k(n-k)! =1,quadlim_nrightarrow infty left(1-fracmnright)^-k =1,quad lim_nrightarrow infty left(1-fracmnright)^n =e^-m
endalignedendequation
$$
Thus, we have the final result which is equal to the formula for the Poisson distribution.
$$
=fracm^k e^-mk!
$$
In all these steps, what I don't understand is the following limit:
$$
lim_nrightarrow infty fracn!n^k(n-k)! =1
$$
limits factorial
asked 2 days ago
billyandrbillyandr
237
$endgroup$
I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable.
Consider the binomial distribution:
$$
beginequationbeginaligned
P(X=k) &=binom n k p^k (1-p)^n-k\
&=fracn!k!(n-k)! p^k (1-p)^n-k
endalignedendequation
$$
Substitute $m=np$ , or $p=fracmn$ :
$$
beginequationbeginaligned
P(X=k) &=fracn!k!(n-k)! left(fracmnright)^k left(1-fracmnright)^n-k\
&=fracn!k!(n-k)! fracm^kn^k left(1-fracmnright)^nleft(1-fracmnright)^-k
endalignedendequation
$$
Slightly rearrange
$$
beginequationbeginaligned
&=fracn!n^k(n-k)! left(1-fracmnright)^-kfracm^kk!left(1-fracmnright)^n
endalignedendequation
$$
Note that
$$
beginequationbeginaligned
& lim_nrightarrow infty fracn!n^k(n-k)! =1,quadlim_nrightarrow infty left(1-fracmnright)^-k =1,quad lim_nrightarrow infty left(1-fracmnright)^n =e^-m
endalignedendequation
$$
Thus, we have the final result which is equal to the formula for the Poisson distribution.
$$
=fracm^k e^-mk!
$$
In all these steps, what I don't understand is the following limit:
$$
lim_nrightarrow infty fracn!n^k(n-k)! =1
$$
limits factorial
limits factorial
asked 2 days ago
billyandrbillyandr
237
asked 2 days ago
billyandrbillyandr
237
edited yesterday
billyandr
asked 2 days ago
billyandrbillyandr
237
asked 2 days ago
billyandrbillyandr
237
asked 2 days ago
billyandrbillyandr
237
237
$begingroup$
There are several posts about this: Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1, Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$, Proof that $limlimits_h to infty frach!h^k(h-k)!=1$ for any $k$, Limits involing Factorials $lim_Ntoinfty fracN!(N-k)!N^k$
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
I found the posts in the above comment using Approach0. For some useful tips on searching here see: How to search on this site?
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
Some of the other posts treating the same question painfully lack details and context. Maybe you'd want to put them on hold or close them.
$endgroup$
– billyandr
yesterday
$begingroup$
billyandr: If you actually have a look at those links, you can see that two of those posts are closed (as duplicates) now. Let me also say that the fact that you have added some more context to your question is certainly appreciated. (After all, that's what lead to reopening.)
$endgroup$
– Martin Sleziak
yesterday
add a comment |
$begingroup$
There are several posts about this: Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1, Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$, Proof that $limlimits_h to infty frach!h^k(h-k)!=1$ for any $k$, Limits involing Factorials $lim_Ntoinfty fracN!(N-k)!N^k$
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
I found the posts in the above comment using Approach0. For some useful tips on searching here see: How to search on this site?
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
Some of the other posts treating the same question painfully lack details and context. Maybe you'd want to put them on hold or close them.
$endgroup$
– billyandr
yesterday
$begingroup$
billyandr: If you actually have a look at those links, you can see that two of those posts are closed (as duplicates) now. Let me also say that the fact that you have added some more context to your question is certainly appreciated. (After all, that's what lead to reopening.)
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
There are several posts about this: Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1, Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$, Proof that $limlimits_h to infty frach!h^k(h-k)!=1$ for any $k$, Limits involing Factorials $lim_Ntoinfty fracN!(N-k)!N^k$
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
There are several posts about this: Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1, Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$, Proof that $limlimits_h to infty frach!h^k(h-k)!=1$ for any $k$, Limits involing Factorials $lim_Ntoinfty fracN!(N-k)!N^k$
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
I found the posts in the above comment using Approach0. For some useful tips on searching here see: How to search on this site?
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
I found the posts in the above comment using Approach0. For some useful tips on searching here see: How to search on this site?
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
Some of the other posts treating the same question painfully lack details and context. Maybe you'd want to put them on hold or close them.
$endgroup$
– billyandr
yesterday
$begingroup$
Some of the other posts treating the same question painfully lack details and context. Maybe you'd want to put them on hold or close them.
$endgroup$
– billyandr
yesterday
$begingroup$
billyandr: If you actually have a look at those links, you can see that two of those posts are closed (as duplicates) now. Let me also say that the fact that you have added some more context to your question is certainly appreciated. (After all, that's what lead to reopening.)
$endgroup$
– Martin Sleziak
yesterday
$begingroup$
billyandr: If you actually have a look at those links, you can see that two of those posts are closed (as duplicates) now. Let me also say that the fact that you have added some more context to your question is certainly appreciated. (After all, that's what lead to reopening.)
$endgroup$
– Martin Sleziak
yesterday
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
It is rather obvious if you cancel the factorials:
$$fracn!n^k(n-k)! =fracoverbracen(n-1)cdots (n-k+1)^k; factorsn^k= 1cdot left(1-frac1nright)cdots left(1-frack-1nright)stackreln to inftylongrightarrow 1$$
answered 2 days ago
trancelocationtrancelocation
14.1k1829
$endgroup$
$begingroup$
Thank you so much. I didn't know it was right there under my eyes.
$endgroup$
– billyandr
2 days ago
$begingroup$
You are welcome. This "not seeing the obvious" just happens once in a while, I think, to all who do maths. So, it is good to have a math platform like this one. :-)
$endgroup$
– trancelocation
2 days ago
add a comment |
$begingroup$
$$a_n=fracn!n^k(n-k)! implies log(a_n)=log(n!)-k log(n)-log((n-k)!)$$
Use Stirling approximation and continue with Taylor series to get
$$log(a_n)=frack(1-k)2 n+Oleft(frac1n^2right)$$ Continue with Taylor
$$a_n=e^log(a_n)=1+frack(1-k)2 n+Oleft(frac1n^2right)$$
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
$endgroup$
1
$begingroup$
This has already a slight touch of overkill, hasn't it? :-)
$endgroup$
– trancelocation
2 days ago
$begingroup$
@trancelocation. You are totally right for the limit. One of my manias is to always look at the approach to the limit. Have a look at matheducators.stackexchange.com/questions/8339/… . Cheers :-)
$endgroup$
– Claude Leibovici
2 days ago
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
$begingroup$
It is rather obvious if you cancel the factorials:
$$fracn!n^k(n-k)! =fracoverbracen(n-1)cdots (n-k+1)^k; factorsn^k= 1cdot left(1-frac1nright)cdots left(1-frack-1nright)stackreln to inftylongrightarrow 1$$
answered 2 days ago
trancelocationtrancelocation
14.1k1829
$endgroup$
$begingroup$
Thank you so much. I didn't know it was right there under my eyes.
$endgroup$
– billyandr
2 days ago
$begingroup$
You are welcome. This "not seeing the obvious" just happens once in a while, I think, to all who do maths. So, it is good to have a math platform like this one. :-)
$endgroup$
– trancelocation
2 days ago
add a comment |
$begingroup$
It is rather obvious if you cancel the factorials:
$$fracn!n^k(n-k)! =fracoverbracen(n-1)cdots (n-k+1)^k; factorsn^k= 1cdot left(1-frac1nright)cdots left(1-frack-1nright)stackreln to inftylongrightarrow 1$$
answered 2 days ago
trancelocationtrancelocation
14.1k1829
$endgroup$
$begingroup$
Thank you so much. I didn't know it was right there under my eyes.
$endgroup$
– billyandr
2 days ago
$begingroup$
You are welcome. This "not seeing the obvious" just happens once in a while, I think, to all who do maths. So, it is good to have a math platform like this one. :-)
$endgroup$
– trancelocation
2 days ago
add a comment |
$begingroup$
It is rather obvious if you cancel the factorials:
$$fracn!n^k(n-k)! =fracoverbracen(n-1)cdots (n-k+1)^k; factorsn^k= 1cdot left(1-frac1nright)cdots left(1-frack-1nright)stackreln to inftylongrightarrow 1$$
answered 2 days ago
trancelocationtrancelocation
14.1k1829
$endgroup$
It is rather obvious if you cancel the factorials:
$$fracn!n^k(n-k)! =fracoverbracen(n-1)cdots (n-k+1)^k; factorsn^k= 1cdot left(1-frac1nright)cdots left(1-frack-1nright)stackreln to inftylongrightarrow 1$$
answered 2 days ago
trancelocationtrancelocation
14.1k1829
answered 2 days ago
trancelocationtrancelocation
14.1k1829
answered 2 days ago
trancelocationtrancelocation
14.1k1829
answered 2 days ago
trancelocationtrancelocation
14.1k1829
14.1k1829
$begingroup$
Thank you so much. I didn't know it was right there under my eyes.
$endgroup$
– billyandr
2 days ago
$begingroup$
You are welcome. This "not seeing the obvious" just happens once in a while, I think, to all who do maths. So, it is good to have a math platform like this one. :-)
$endgroup$
– trancelocation
2 days ago
add a comment |
$begingroup$
Thank you so much. I didn't know it was right there under my eyes.
$endgroup$
– billyandr
2 days ago
$begingroup$
You are welcome. This "not seeing the obvious" just happens once in a while, I think, to all who do maths. So, it is good to have a math platform like this one. :-)
$endgroup$
– trancelocation
2 days ago
$begingroup$
Thank you so much. I didn't know it was right there under my eyes.
$endgroup$
– billyandr
2 days ago
$begingroup$
Thank you so much. I didn't know it was right there under my eyes.
$endgroup$
– billyandr
2 days ago
$begingroup$
You are welcome. This "not seeing the obvious" just happens once in a while, I think, to all who do maths. So, it is good to have a math platform like this one. :-)
$endgroup$
– trancelocation
2 days ago
$begingroup$
You are welcome. This "not seeing the obvious" just happens once in a while, I think, to all who do maths. So, it is good to have a math platform like this one. :-)
$endgroup$
– trancelocation
2 days ago
add a comment |
$begingroup$
$$a_n=fracn!n^k(n-k)! implies log(a_n)=log(n!)-k log(n)-log((n-k)!)$$
Use Stirling approximation and continue with Taylor series to get
$$log(a_n)=frack(1-k)2 n+Oleft(frac1n^2right)$$ Continue with Taylor
$$a_n=e^log(a_n)=1+frack(1-k)2 n+Oleft(frac1n^2right)$$
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
$endgroup$
1
$begingroup$
This has already a slight touch of overkill, hasn't it? :-)
$endgroup$
– trancelocation
2 days ago
$begingroup$
@trancelocation. You are totally right for the limit. One of my manias is to always look at the approach to the limit. Have a look at matheducators.stackexchange.com/questions/8339/… . Cheers :-)
$endgroup$
– Claude Leibovici
2 days ago
add a comment |
$begingroup$
$$a_n=fracn!n^k(n-k)! implies log(a_n)=log(n!)-k log(n)-log((n-k)!)$$
Use Stirling approximation and continue with Taylor series to get
$$log(a_n)=frack(1-k)2 n+Oleft(frac1n^2right)$$ Continue with Taylor
$$a_n=e^log(a_n)=1+frack(1-k)2 n+Oleft(frac1n^2right)$$
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
$endgroup$
1
$begingroup$
This has already a slight touch of overkill, hasn't it? :-)
$endgroup$
– trancelocation
2 days ago
$begingroup$
@trancelocation. You are totally right for the limit. One of my manias is to always look at the approach to the limit. Have a look at matheducators.stackexchange.com/questions/8339/… . Cheers :-)
$endgroup$
– Claude Leibovici
2 days ago
add a comment |
$begingroup$
$$a_n=fracn!n^k(n-k)! implies log(a_n)=log(n!)-k log(n)-log((n-k)!)$$
Use Stirling approximation and continue with Taylor series to get
$$log(a_n)=frack(1-k)2 n+Oleft(frac1n^2right)$$ Continue with Taylor
$$a_n=e^log(a_n)=1+frack(1-k)2 n+Oleft(frac1n^2right)$$
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
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$$a_n=fracn!n^k(n-k)! implies log(a_n)=log(n!)-k log(n)-log((n-k)!)$$
Use Stirling approximation and continue with Taylor series to get
$$log(a_n)=frack(1-k)2 n+Oleft(frac1n^2right)$$ Continue with Taylor
$$a_n=e^log(a_n)=1+frack(1-k)2 n+Oleft(frac1n^2right)$$
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
answered 2 days ago
Claude LeiboviciClaude Leibovici
126k1158135
126k1158135
1
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This has already a slight touch of overkill, hasn't it? :-)
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– trancelocation
2 days ago
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@trancelocation. You are totally right for the limit. One of my manias is to always look at the approach to the limit. Have a look at matheducators.stackexchange.com/questions/8339/… . Cheers :-)
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– Claude Leibovici
2 days ago
add a comment |
1
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This has already a slight touch of overkill, hasn't it? :-)
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– trancelocation
2 days ago
$begingroup$
@trancelocation. You are totally right for the limit. One of my manias is to always look at the approach to the limit. Have a look at matheducators.stackexchange.com/questions/8339/… . Cheers :-)
$endgroup$
– Claude Leibovici
2 days ago
1
1
$begingroup$
This has already a slight touch of overkill, hasn't it? :-)
$endgroup$
– trancelocation
2 days ago
$begingroup$
This has already a slight touch of overkill, hasn't it? :-)
$endgroup$
– trancelocation
2 days ago
$begingroup$
@trancelocation. You are totally right for the limit. One of my manias is to always look at the approach to the limit. Have a look at matheducators.stackexchange.com/questions/8339/… . Cheers :-)
$endgroup$
– Claude Leibovici
2 days ago
$begingroup$
@trancelocation. You are totally right for the limit. One of my manias is to always look at the approach to the limit. Have a look at matheducators.stackexchange.com/questions/8339/… . Cheers :-)
$endgroup$
– Claude Leibovici
2 days ago
add a comment |
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There are several posts about this: Why does $lim_ntoinfty fracn!(n-k)!n^k$ equal 1, Finding limit of sequence: $lim _n to infty fracn!n^k(n-k)!=1$, Proof that $limlimits_h to infty frach!h^k(h-k)!=1$ for any $k$, Limits involing Factorials $lim_Ntoinfty fracN!(N-k)!N^k$
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– Martin Sleziak
yesterday
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I found the posts in the above comment using Approach0. For some useful tips on searching here see: How to search on this site?
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– Martin Sleziak
yesterday
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Some of the other posts treating the same question painfully lack details and context. Maybe you'd want to put them on hold or close them.
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– billyandr
yesterday
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billyandr: If you actually have a look at those links, you can see that two of those posts are closed (as duplicates) now. Let me also say that the fact that you have added some more context to your question is certainly appreciated. (After all, that's what lead to reopening.)
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– Martin Sleziak
yesterday