How to prove the limit $lim_{krightarrowinfty}lambda^{k-j}binom{k}{k-j}=0,; k>j,$ and $kgt j$
0
$begingroup$
I find difficulty proving the following limit:
Suppose $|lambda|<1$,then
$$
lim_{krightarrowinfty}lambda^{k-j}binom{k}{k-j}=0,quad k>j.
$$
When I try to prove that the $k$th Jordan matrix converges to $0$ ... (Please see this earlier question.)
linear-algebra limits
edited Jan 20 at 20:01
jordan_glen
1
asked Jan 20 at 19:37
whereamIwhereamI
322115
$endgroup$
|
show 2 more comments
0
$begingroup$
I find difficulty proving the following limit:
Suppose $|lambda|<1$,then
$$
lim_{krightarrowinfty}lambda^{k-j}binom{k}{k-j}=0,quad k>j.
$$
When I try to prove that the $k$th Jordan matrix converges to $0$ ... (Please see this earlier question.)
linear-algebra limits
edited Jan 20 at 20:01
jordan_glen
1
asked Jan 20 at 19:37
whereamIwhereamI
322115
$endgroup$
1
$begingroup$
Hont. For each fixed $j$, the binomial coefficient $binom{k}{k-j}$ is a polynomial of $k$.
$endgroup$
– Sangchul Lee
Jan 20 at 19:41
2
$begingroup$
Hant. Stirling's formula may be of use, see en.wikipedia.org/wiki/Stirling%27s_approximation
$endgroup$
– P. Quinton
Jan 20 at 19:48
$begingroup$
@SangchulLee I see it. Thank you very much.
$endgroup$
– whereamI
Jan 20 at 19:49
$begingroup$
@P.Quinton That's formula is quite useful. Thanks a lot!.
$endgroup$
– whereamI
Jan 20 at 19:53
$begingroup$
What happens when you try to prove that the kth Jordan matrix converges to $0$?
$endgroup$
– jordan_glen
Jan 20 at 19:53
|
show 2 more comments
0
0
0
$begingroup$
I find difficulty proving the following limit:
Suppose $|lambda|<1$,then
$$
lim_{krightarrowinfty}lambda^{k-j}binom{k}{k-j}=0,quad k>j.
$$
When I try to prove that the $k$th Jordan matrix converges to $0$ ... (Please see this earlier question.)
linear-algebra limits
edited Jan 20 at 20:01
jordan_glen
1
asked Jan 20 at 19:37
whereamIwhereamI
322115
$endgroup$
I find difficulty proving the following limit:
Suppose $|lambda|<1$,then
$$
lim_{krightarrowinfty}lambda^{k-j}binom{k}{k-j}=0,quad k>j.
$$
When I try to prove that the $k$th Jordan matrix converges to $0$ ... (Please see this earlier question.)
linear-algebra limits
linear-algebra limits
edited Jan 20 at 20:01
jordan_glen
1
asked Jan 20 at 19:37
whereamIwhereamI
322115
edited Jan 20 at 20:01
jordan_glen
1
asked Jan 20 at 19:37
whereamIwhereamI
322115
edited Jan 20 at 20:01
jordan_glen
1
edited Jan 20 at 20:01
jordan_glen
1
edited Jan 20 at 20:01
jordan_glen
1
1
asked Jan 20 at 19:37
whereamIwhereamI
322115
asked Jan 20 at 19:37
whereamIwhereamI
322115
asked Jan 20 at 19:37
whereamIwhereamI
322115
322115
1
$begingroup$
Hont. For each fixed $j$, the binomial coefficient $binom{k}{k-j}$ is a polynomial of $k$.
$endgroup$
– Sangchul Lee
Jan 20 at 19:41
2
$begingroup$
Hant. Stirling's formula may be of use, see en.wikipedia.org/wiki/Stirling%27s_approximation
$endgroup$
– P. Quinton
Jan 20 at 19:48
$begingroup$
@SangchulLee I see it. Thank you very much.
$endgroup$
– whereamI
Jan 20 at 19:49
$begingroup$
@P.Quinton That's formula is quite useful. Thanks a lot!.
$endgroup$
– whereamI
Jan 20 at 19:53
$begingroup$
What happens when you try to prove that the kth Jordan matrix converges to $0$?
$endgroup$
– jordan_glen
Jan 20 at 19:53
|
show 2 more comments
1
$begingroup$
Hont. For each fixed $j$, the binomial coefficient $binom{k}{k-j}$ is a polynomial of $k$.
$endgroup$
– Sangchul Lee
Jan 20 at 19:41
2
$begingroup$
Hant. Stirling's formula may be of use, see en.wikipedia.org/wiki/Stirling%27s_approximation
$endgroup$
– P. Quinton
Jan 20 at 19:48
$begingroup$
@SangchulLee I see it. Thank you very much.
$endgroup$
– whereamI
Jan 20 at 19:49
$begingroup$
@P.Quinton That's formula is quite useful. Thanks a lot!.
$endgroup$
– whereamI
Jan 20 at 19:53
$begingroup$
What happens when you try to prove that the kth Jordan matrix converges to $0$?
$endgroup$
– jordan_glen
Jan 20 at 19:53
1
1
$begingroup$
Hont. For each fixed $j$, the binomial coefficient $binom{k}{k-j}$ is a polynomial of $k$.
$endgroup$
– Sangchul Lee
Jan 20 at 19:41
$begingroup$
Hont. For each fixed $j$, the binomial coefficient $binom{k}{k-j}$ is a polynomial of $k$.
$endgroup$
– Sangchul Lee
Jan 20 at 19:41
2
2
$begingroup$
Hant. Stirling's formula may be of use, see en.wikipedia.org/wiki/Stirling%27s_approximation
$endgroup$
– P. Quinton
Jan 20 at 19:48
$begingroup$
Hant. Stirling's formula may be of use, see en.wikipedia.org/wiki/Stirling%27s_approximation
$endgroup$
– P. Quinton
Jan 20 at 19:48
$begingroup$
@SangchulLee I see it. Thank you very much.
$endgroup$
– whereamI
Jan 20 at 19:49
$begingroup$
@SangchulLee I see it. Thank you very much.
$endgroup$
– whereamI
Jan 20 at 19:49
$begingroup$
@P.Quinton That's formula is quite useful. Thanks a lot!.
$endgroup$
– whereamI
Jan 20 at 19:53
$begingroup$
@P.Quinton That's formula is quite useful. Thanks a lot!.
$endgroup$
– whereamI
Jan 20 at 19:53
$begingroup$
What happens when you try to prove that the kth Jordan matrix converges to $0$?
$endgroup$
– jordan_glen
Jan 20 at 19:53
$begingroup$
What happens when you try to prove that the kth Jordan matrix converges to $0$?
$endgroup$
– jordan_glen
Jan 20 at 19:53
|
show 2 more comments
0
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1
$begingroup$
Hont. For each fixed $j$, the binomial coefficient $binom{k}{k-j}$ is a polynomial of $k$.
$endgroup$
– Sangchul Lee
Jan 20 at 19:41
2
$begingroup$
Hant. Stirling's formula may be of use, see en.wikipedia.org/wiki/Stirling%27s_approximation
$endgroup$
– P. Quinton
Jan 20 at 19:48
$begingroup$
@SangchulLee I see it. Thank you very much.
$endgroup$
– whereamI
Jan 20 at 19:49
$begingroup$
@P.Quinton That's formula is quite useful. Thanks a lot!.
$endgroup$
– whereamI
Jan 20 at 19:53
$begingroup$
What happens when you try to prove that the kth Jordan matrix converges to $0$?
$endgroup$
– jordan_glen
Jan 20 at 19:53