Mixing
determine whether the sequence $f_{n}(x) = n x e^{-nx}$ converges uniformly on the set [0,1]
$begingroup$
Determine whether the sequence $f_{n}(x) = n x e^{-nx}$ converges uniformly on the set [0,1].
My trial:
The pointwise limit is zero by L`hopitals rule , but then what how can I find N so that $|nxe^{-nx}| < epsilon$
calculus sequences-and-series
asked Jan 27 at 18:03
hopefullyhopefully
269114
$endgroup$
add a comment |
$begingroup$
Determine whether the sequence $f_{n}(x) = n x e^{-nx}$ converges uniformly on the set [0,1].
My trial:
The pointwise limit is zero by L`hopitals rule , but then what how can I find N so that $|nxe^{-nx}| < epsilon$
calculus sequences-and-series
asked Jan 27 at 18:03
hopefullyhopefully
269114
$endgroup$
add a comment |
$begingroup$
Determine whether the sequence $f_{n}(x) = n x e^{-nx}$ converges uniformly on the set [0,1].
My trial:
The pointwise limit is zero by L`hopitals rule , but then what how can I find N so that $|nxe^{-nx}| < epsilon$
calculus sequences-and-series
asked Jan 27 at 18:03
hopefullyhopefully
269114
$endgroup$
Determine whether the sequence $f_{n}(x) = n x e^{-nx}$ converges uniformly on the set [0,1].
My trial:
The pointwise limit is zero by L`hopitals rule , but then what how can I find N so that $|nxe^{-nx}| < epsilon$
calculus sequences-and-series
calculus sequences-and-series
asked Jan 27 at 18:03
hopefullyhopefully
269114
asked Jan 27 at 18:03
hopefullyhopefully
269114
asked Jan 27 at 18:03
hopefullyhopefully
269114
asked Jan 27 at 18:03
hopefullyhopefully
269114
asked Jan 27 at 18:03
hopefullyhopefully
269114
269114
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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Suppose $f_n(x)$ converges uniformly to zero function. Then given $varepsilon gt 0 $, there exists $n_0in Bbb N $ such that $|f_n(x)-0|<varepsilon$, for all $ngeq n_0$ and $xin [0,1]$. Note that $$f_nleft(frac1nright)=frac1e .$$ Can you now find a suitable $varepsilon$ which contradicts our assumption?
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
$endgroup$
$begingroup$
take $epsilon = 0.1$ ...... is it correct?
$endgroup$
– hopefully
Jan 27 at 18:28
$begingroup$
The supremum is $1/n$ and not zero
$endgroup$
– hopefully
Jan 27 at 21:59
1
$begingroup$
@hopefully Yes, $0.1$ will work.
$endgroup$
– Thomas Shelby
Jan 28 at 1:14
add a comment |
$begingroup$
Hint: Find the maximum value of $f_n(x)$ on $[0,1]$. Does it approach $0$ as $ntoinfty$?
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
$endgroup$
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
$begingroup$
Suppose $f_n(x)$ converges uniformly to zero function. Then given $varepsilon gt 0 $, there exists $n_0in Bbb N $ such that $|f_n(x)-0|<varepsilon$, for all $ngeq n_0$ and $xin [0,1]$. Note that $$f_nleft(frac1nright)=frac1e .$$ Can you now find a suitable $varepsilon$ which contradicts our assumption?
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
$endgroup$
$begingroup$
take $epsilon = 0.1$ ...... is it correct?
$endgroup$
– hopefully
Jan 27 at 18:28
$begingroup$
The supremum is $1/n$ and not zero
$endgroup$
– hopefully
Jan 27 at 21:59
1
$begingroup$
@hopefully Yes, $0.1$ will work.
$endgroup$
– Thomas Shelby
Jan 28 at 1:14
add a comment |
$begingroup$
Suppose $f_n(x)$ converges uniformly to zero function. Then given $varepsilon gt 0 $, there exists $n_0in Bbb N $ such that $|f_n(x)-0|<varepsilon$, for all $ngeq n_0$ and $xin [0,1]$. Note that $$f_nleft(frac1nright)=frac1e .$$ Can you now find a suitable $varepsilon$ which contradicts our assumption?
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
$endgroup$
$begingroup$
take $epsilon = 0.1$ ...... is it correct?
$endgroup$
– hopefully
Jan 27 at 18:28
$begingroup$
The supremum is $1/n$ and not zero
$endgroup$
– hopefully
Jan 27 at 21:59
1
$begingroup$
@hopefully Yes, $0.1$ will work.
$endgroup$
– Thomas Shelby
Jan 28 at 1:14
add a comment |
$begingroup$
Suppose $f_n(x)$ converges uniformly to zero function. Then given $varepsilon gt 0 $, there exists $n_0in Bbb N $ such that $|f_n(x)-0|<varepsilon$, for all $ngeq n_0$ and $xin [0,1]$. Note that $$f_nleft(frac1nright)=frac1e .$$ Can you now find a suitable $varepsilon$ which contradicts our assumption?
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
$endgroup$
Suppose $f_n(x)$ converges uniformly to zero function. Then given $varepsilon gt 0 $, there exists $n_0in Bbb N $ such that $|f_n(x)-0|<varepsilon$, for all $ngeq n_0$ and $xin [0,1]$. Note that $$f_nleft(frac1nright)=frac1e .$$ Can you now find a suitable $varepsilon$ which contradicts our assumption?
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
edited Jan 27 at 18:22
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
answered Jan 27 at 18:17
Thomas ShelbyThomas Shelby
4,4292726
4,4292726
$begingroup$
take $epsilon = 0.1$ ...... is it correct?
$endgroup$
– hopefully
Jan 27 at 18:28
$begingroup$
The supremum is $1/n$ and not zero
$endgroup$
– hopefully
Jan 27 at 21:59
1
$begingroup$
@hopefully Yes, $0.1$ will work.
$endgroup$
– Thomas Shelby
Jan 28 at 1:14
add a comment |
$begingroup$
take $epsilon = 0.1$ ...... is it correct?
$endgroup$
– hopefully
Jan 27 at 18:28
$begingroup$
The supremum is $1/n$ and not zero
$endgroup$
– hopefully
Jan 27 at 21:59
1
$begingroup$
@hopefully Yes, $0.1$ will work.
$endgroup$
– Thomas Shelby
Jan 28 at 1:14
$begingroup$
take $epsilon = 0.1$ ...... is it correct?
$endgroup$
– hopefully
Jan 27 at 18:28
$begingroup$
take $epsilon = 0.1$ ...... is it correct?
$endgroup$
– hopefully
Jan 27 at 18:28
$begingroup$
The supremum is $1/n$ and not zero
$endgroup$
– hopefully
Jan 27 at 21:59
$begingroup$
The supremum is $1/n$ and not zero
$endgroup$
– hopefully
Jan 27 at 21:59
1
1
$begingroup$
@hopefully Yes, $0.1$ will work.
$endgroup$
– Thomas Shelby
Jan 28 at 1:14
$begingroup$
@hopefully Yes, $0.1$ will work.
$endgroup$
– Thomas Shelby
Jan 28 at 1:14
add a comment |
$begingroup$
Hint: Find the maximum value of $f_n(x)$ on $[0,1]$. Does it approach $0$ as $ntoinfty$?
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
$endgroup$
add a comment |
$begingroup$
Hint: Find the maximum value of $f_n(x)$ on $[0,1]$. Does it approach $0$ as $ntoinfty$?
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
$endgroup$
add a comment |
$begingroup$
Hint: Find the maximum value of $f_n(x)$ on $[0,1]$. Does it approach $0$ as $ntoinfty$?
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
$endgroup$
Hint: Find the maximum value of $f_n(x)$ on $[0,1]$. Does it approach $0$ as $ntoinfty$?
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
edited Jan 27 at 18:24
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
answered Jan 27 at 18:05
Eclipse SunEclipse Sun
7,8401438
7,8401438
add a comment |
add a comment |
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