Proving Uniform Convergence of a Series of Functions
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I'm working on the following problem:
I've observed that division of zero will occur in a single term of the series if a natural number is selected, which is why that constraint is given.
I've computed a few terms manually for x=1.1, and I can see that the terms' absolute values get smaller quite rapidly.
So, it does appear that the series will converge. My tools to prove this thus far are the M-test and the Cauchy Condition, but I'm unsure if either is suitable in this case. I was also thinking about finding another series of functions that also converges, but has larger terms than the one in this series of functions.
Any assistance would be appreciated. Thank you!
real-analysis sequences-and-series uniform-convergence
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up vote
1
down vote
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I'm working on the following problem:
I've observed that division of zero will occur in a single term of the series if a natural number is selected, which is why that constraint is given.
I've computed a few terms manually for x=1.1, and I can see that the terms' absolute values get smaller quite rapidly.
So, it does appear that the series will converge. My tools to prove this thus far are the M-test and the Cauchy Condition, but I'm unsure if either is suitable in this case. I was also thinking about finding another series of functions that also converges, but has larger terms than the one in this series of functions.
Any assistance would be appreciated. Thank you!
real-analysis sequences-and-series uniform-convergence
1
Hint: M-test. For all sufficiently large $n$, $$left|frac{1}{n(x-n)} right| leqslant frac{1}{n(n-b)}$$
– RRL
Nov 26 at 3:17
Thanks! I'll work getting an M-test to work with that inequality.
– Justin
Nov 26 at 20:06
I spoke with my Professor, and he agreed with you, although he mentioned that I'd need to use cases, as TheD0ubleT did.
– Justin
Nov 28 at 4:40
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm working on the following problem:
I've observed that division of zero will occur in a single term of the series if a natural number is selected, which is why that constraint is given.
I've computed a few terms manually for x=1.1, and I can see that the terms' absolute values get smaller quite rapidly.
So, it does appear that the series will converge. My tools to prove this thus far are the M-test and the Cauchy Condition, but I'm unsure if either is suitable in this case. I was also thinking about finding another series of functions that also converges, but has larger terms than the one in this series of functions.
Any assistance would be appreciated. Thank you!
real-analysis sequences-and-series uniform-convergence
I'm working on the following problem:
I've observed that division of zero will occur in a single term of the series if a natural number is selected, which is why that constraint is given.
I've computed a few terms manually for x=1.1, and I can see that the terms' absolute values get smaller quite rapidly.
So, it does appear that the series will converge. My tools to prove this thus far are the M-test and the Cauchy Condition, but I'm unsure if either is suitable in this case. I was also thinking about finding another series of functions that also converges, but has larger terms than the one in this series of functions.
Any assistance would be appreciated. Thank you!
real-analysis sequences-and-series uniform-convergence
real-analysis sequences-and-series uniform-convergence
asked Nov 26 at 1:43
Justin
316
316
1
Hint: M-test. For all sufficiently large $n$, $$left|frac{1}{n(x-n)} right| leqslant frac{1}{n(n-b)}$$
– RRL
Nov 26 at 3:17
Thanks! I'll work getting an M-test to work with that inequality.
– Justin
Nov 26 at 20:06
I spoke with my Professor, and he agreed with you, although he mentioned that I'd need to use cases, as TheD0ubleT did.
– Justin
Nov 28 at 4:40
add a comment |
1
Hint: M-test. For all sufficiently large $n$, $$left|frac{1}{n(x-n)} right| leqslant frac{1}{n(n-b)}$$
– RRL
Nov 26 at 3:17
Thanks! I'll work getting an M-test to work with that inequality.
– Justin
Nov 26 at 20:06
I spoke with my Professor, and he agreed with you, although he mentioned that I'd need to use cases, as TheD0ubleT did.
– Justin
Nov 28 at 4:40
1
1
Hint: M-test. For all sufficiently large $n$, $$left|frac{1}{n(x-n)} right| leqslant frac{1}{n(n-b)}$$
– RRL
Nov 26 at 3:17
Hint: M-test. For all sufficiently large $n$, $$left|frac{1}{n(x-n)} right| leqslant frac{1}{n(n-b)}$$
– RRL
Nov 26 at 3:17
Thanks! I'll work getting an M-test to work with that inequality.
– Justin
Nov 26 at 20:06
Thanks! I'll work getting an M-test to work with that inequality.
– Justin
Nov 26 at 20:06
I spoke with my Professor, and he agreed with you, although he mentioned that I'd need to use cases, as TheD0ubleT did.
– Justin
Nov 28 at 4:40
I spoke with my Professor, and he agreed with you, although he mentioned that I'd need to use cases, as TheD0ubleT did.
– Justin
Nov 28 at 4:40
add a comment |
2 Answers
2
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up vote
1
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accepted
A variant of a preceding answer, since it's almost impossible to do something extremely different:
For $nin Bbb N$ let $g_n=sum_{j=n}^{infty}1/j^2.$ We have $lim_{nto infty}g_n=0.$
Let $cin Bbb N$ with $cgeqmax (|a|,|b|).$ For $c<jin Bbb N$ and for any $xin [a,b]$ we have $$|j(x-j)|=j|j-x|geq j(|j|-|x|)=j(j-|x|)geq j(j-c)geq (j-c)^2.$$
So if $c<nin Bbb N$ then $$sup_{nleq n_1leq n_2}|sum_{j=n_1}^{n_2}1/j(x-j)|leq sum_{j=n}^{infty}|1/j(x-j)|leq$$ $$leq sum_{j=n}^{infty}1/(j-c)^2=g_{n-c}.$$
And $g_{n-c}to 0$ as $nto infty.$ So the series represented by $f$ satisfies the Cauchy Condition for convergence at each $xin [a,b]$ so it converges to some $f(x).$
And for $n>c$ we have $$lim_{c<nto infty}sup_{xin [a,b]}|f(x)-sum_{j=1}^n 1/j(x-j)|leq lim_{c<nto infty}g_{1+n-c}=0$$ so the convergence is uniform.
Thank you for the answer.
– Justin
Nov 28 at 17:47
add a comment |
up vote
2
down vote
You can build a series $u_n$ which is always bigger than $|frac{1}{n(x-n)}|$
We know that $exists ninBbb{N}, n<a<b<n+1$ because there are no natural numbers in $[a,b]$
Thus, $forall n in Bbb{N^*}, forall xin[a,b], |frac{1}{n(x-n)}|le {begin{matrix}frac{1}{n(a-n)} if n<a \ frac{1}{n(n-b)} if n>b end{matrix} = u_n$
The series we just created always converges because the series is always positive and $u_nsimfrac{1}{n^2}$
$sim$ means "equivalent". It's an asymptotic analysis of a series or function Wiki article here
Thanks for the answer. I believe you, that the series you built is convergent, but is there a more rigorous way to show that it converges other than "it converges because it approximates one over n-squared?"
– Justin
Nov 28 at 4:38
It is rigorous, because $u_n sim frac {1}{n^2}$ implies that $$exists n_0inBbb{N}, forall n > n_0, frac 12*frac {1}{n^2}le u_nle frac 32*frac {1}{n^2}$$ Thus if $sum frac {1}{n^2}$ converges then so does $sum u_n$
– TheD0ubleT
Nov 28 at 8:16
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
A variant of a preceding answer, since it's almost impossible to do something extremely different:
For $nin Bbb N$ let $g_n=sum_{j=n}^{infty}1/j^2.$ We have $lim_{nto infty}g_n=0.$
Let $cin Bbb N$ with $cgeqmax (|a|,|b|).$ For $c<jin Bbb N$ and for any $xin [a,b]$ we have $$|j(x-j)|=j|j-x|geq j(|j|-|x|)=j(j-|x|)geq j(j-c)geq (j-c)^2.$$
So if $c<nin Bbb N$ then $$sup_{nleq n_1leq n_2}|sum_{j=n_1}^{n_2}1/j(x-j)|leq sum_{j=n}^{infty}|1/j(x-j)|leq$$ $$leq sum_{j=n}^{infty}1/(j-c)^2=g_{n-c}.$$
And $g_{n-c}to 0$ as $nto infty.$ So the series represented by $f$ satisfies the Cauchy Condition for convergence at each $xin [a,b]$ so it converges to some $f(x).$
And for $n>c$ we have $$lim_{c<nto infty}sup_{xin [a,b]}|f(x)-sum_{j=1}^n 1/j(x-j)|leq lim_{c<nto infty}g_{1+n-c}=0$$ so the convergence is uniform.
Thank you for the answer.
– Justin
Nov 28 at 17:47
add a comment |
up vote
1
down vote
accepted
A variant of a preceding answer, since it's almost impossible to do something extremely different:
For $nin Bbb N$ let $g_n=sum_{j=n}^{infty}1/j^2.$ We have $lim_{nto infty}g_n=0.$
Let $cin Bbb N$ with $cgeqmax (|a|,|b|).$ For $c<jin Bbb N$ and for any $xin [a,b]$ we have $$|j(x-j)|=j|j-x|geq j(|j|-|x|)=j(j-|x|)geq j(j-c)geq (j-c)^2.$$
So if $c<nin Bbb N$ then $$sup_{nleq n_1leq n_2}|sum_{j=n_1}^{n_2}1/j(x-j)|leq sum_{j=n}^{infty}|1/j(x-j)|leq$$ $$leq sum_{j=n}^{infty}1/(j-c)^2=g_{n-c}.$$
And $g_{n-c}to 0$ as $nto infty.$ So the series represented by $f$ satisfies the Cauchy Condition for convergence at each $xin [a,b]$ so it converges to some $f(x).$
And for $n>c$ we have $$lim_{c<nto infty}sup_{xin [a,b]}|f(x)-sum_{j=1}^n 1/j(x-j)|leq lim_{c<nto infty}g_{1+n-c}=0$$ so the convergence is uniform.
Thank you for the answer.
– Justin
Nov 28 at 17:47
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
A variant of a preceding answer, since it's almost impossible to do something extremely different:
For $nin Bbb N$ let $g_n=sum_{j=n}^{infty}1/j^2.$ We have $lim_{nto infty}g_n=0.$
Let $cin Bbb N$ with $cgeqmax (|a|,|b|).$ For $c<jin Bbb N$ and for any $xin [a,b]$ we have $$|j(x-j)|=j|j-x|geq j(|j|-|x|)=j(j-|x|)geq j(j-c)geq (j-c)^2.$$
So if $c<nin Bbb N$ then $$sup_{nleq n_1leq n_2}|sum_{j=n_1}^{n_2}1/j(x-j)|leq sum_{j=n}^{infty}|1/j(x-j)|leq$$ $$leq sum_{j=n}^{infty}1/(j-c)^2=g_{n-c}.$$
And $g_{n-c}to 0$ as $nto infty.$ So the series represented by $f$ satisfies the Cauchy Condition for convergence at each $xin [a,b]$ so it converges to some $f(x).$
And for $n>c$ we have $$lim_{c<nto infty}sup_{xin [a,b]}|f(x)-sum_{j=1}^n 1/j(x-j)|leq lim_{c<nto infty}g_{1+n-c}=0$$ so the convergence is uniform.
A variant of a preceding answer, since it's almost impossible to do something extremely different:
For $nin Bbb N$ let $g_n=sum_{j=n}^{infty}1/j^2.$ We have $lim_{nto infty}g_n=0.$
Let $cin Bbb N$ with $cgeqmax (|a|,|b|).$ For $c<jin Bbb N$ and for any $xin [a,b]$ we have $$|j(x-j)|=j|j-x|geq j(|j|-|x|)=j(j-|x|)geq j(j-c)geq (j-c)^2.$$
So if $c<nin Bbb N$ then $$sup_{nleq n_1leq n_2}|sum_{j=n_1}^{n_2}1/j(x-j)|leq sum_{j=n}^{infty}|1/j(x-j)|leq$$ $$leq sum_{j=n}^{infty}1/(j-c)^2=g_{n-c}.$$
And $g_{n-c}to 0$ as $nto infty.$ So the series represented by $f$ satisfies the Cauchy Condition for convergence at each $xin [a,b]$ so it converges to some $f(x).$
And for $n>c$ we have $$lim_{c<nto infty}sup_{xin [a,b]}|f(x)-sum_{j=1}^n 1/j(x-j)|leq lim_{c<nto infty}g_{1+n-c}=0$$ so the convergence is uniform.
edited Nov 28 at 14:37
answered Nov 28 at 14:18
DanielWainfleet
33.7k31647
33.7k31647
Thank you for the answer.
– Justin
Nov 28 at 17:47
add a comment |
Thank you for the answer.
– Justin
Nov 28 at 17:47
Thank you for the answer.
– Justin
Nov 28 at 17:47
Thank you for the answer.
– Justin
Nov 28 at 17:47
add a comment |
up vote
2
down vote
You can build a series $u_n$ which is always bigger than $|frac{1}{n(x-n)}|$
We know that $exists ninBbb{N}, n<a<b<n+1$ because there are no natural numbers in $[a,b]$
Thus, $forall n in Bbb{N^*}, forall xin[a,b], |frac{1}{n(x-n)}|le {begin{matrix}frac{1}{n(a-n)} if n<a \ frac{1}{n(n-b)} if n>b end{matrix} = u_n$
The series we just created always converges because the series is always positive and $u_nsimfrac{1}{n^2}$
$sim$ means "equivalent". It's an asymptotic analysis of a series or function Wiki article here
Thanks for the answer. I believe you, that the series you built is convergent, but is there a more rigorous way to show that it converges other than "it converges because it approximates one over n-squared?"
– Justin
Nov 28 at 4:38
It is rigorous, because $u_n sim frac {1}{n^2}$ implies that $$exists n_0inBbb{N}, forall n > n_0, frac 12*frac {1}{n^2}le u_nle frac 32*frac {1}{n^2}$$ Thus if $sum frac {1}{n^2}$ converges then so does $sum u_n$
– TheD0ubleT
Nov 28 at 8:16
add a comment |
up vote
2
down vote
You can build a series $u_n$ which is always bigger than $|frac{1}{n(x-n)}|$
We know that $exists ninBbb{N}, n<a<b<n+1$ because there are no natural numbers in $[a,b]$
Thus, $forall n in Bbb{N^*}, forall xin[a,b], |frac{1}{n(x-n)}|le {begin{matrix}frac{1}{n(a-n)} if n<a \ frac{1}{n(n-b)} if n>b end{matrix} = u_n$
The series we just created always converges because the series is always positive and $u_nsimfrac{1}{n^2}$
$sim$ means "equivalent". It's an asymptotic analysis of a series or function Wiki article here
Thanks for the answer. I believe you, that the series you built is convergent, but is there a more rigorous way to show that it converges other than "it converges because it approximates one over n-squared?"
– Justin
Nov 28 at 4:38
It is rigorous, because $u_n sim frac {1}{n^2}$ implies that $$exists n_0inBbb{N}, forall n > n_0, frac 12*frac {1}{n^2}le u_nle frac 32*frac {1}{n^2}$$ Thus if $sum frac {1}{n^2}$ converges then so does $sum u_n$
– TheD0ubleT
Nov 28 at 8:16
add a comment |
up vote
2
down vote
up vote
2
down vote
You can build a series $u_n$ which is always bigger than $|frac{1}{n(x-n)}|$
We know that $exists ninBbb{N}, n<a<b<n+1$ because there are no natural numbers in $[a,b]$
Thus, $forall n in Bbb{N^*}, forall xin[a,b], |frac{1}{n(x-n)}|le {begin{matrix}frac{1}{n(a-n)} if n<a \ frac{1}{n(n-b)} if n>b end{matrix} = u_n$
The series we just created always converges because the series is always positive and $u_nsimfrac{1}{n^2}$
$sim$ means "equivalent". It's an asymptotic analysis of a series or function Wiki article here
You can build a series $u_n$ which is always bigger than $|frac{1}{n(x-n)}|$
We know that $exists ninBbb{N}, n<a<b<n+1$ because there are no natural numbers in $[a,b]$
Thus, $forall n in Bbb{N^*}, forall xin[a,b], |frac{1}{n(x-n)}|le {begin{matrix}frac{1}{n(a-n)} if n<a \ frac{1}{n(n-b)} if n>b end{matrix} = u_n$
The series we just created always converges because the series is always positive and $u_nsimfrac{1}{n^2}$
$sim$ means "equivalent". It's an asymptotic analysis of a series or function Wiki article here
edited Nov 28 at 9:14
answered Nov 26 at 9:45
TheD0ubleT
38118
38118
Thanks for the answer. I believe you, that the series you built is convergent, but is there a more rigorous way to show that it converges other than "it converges because it approximates one over n-squared?"
– Justin
Nov 28 at 4:38
It is rigorous, because $u_n sim frac {1}{n^2}$ implies that $$exists n_0inBbb{N}, forall n > n_0, frac 12*frac {1}{n^2}le u_nle frac 32*frac {1}{n^2}$$ Thus if $sum frac {1}{n^2}$ converges then so does $sum u_n$
– TheD0ubleT
Nov 28 at 8:16
add a comment |
Thanks for the answer. I believe you, that the series you built is convergent, but is there a more rigorous way to show that it converges other than "it converges because it approximates one over n-squared?"
– Justin
Nov 28 at 4:38
It is rigorous, because $u_n sim frac {1}{n^2}$ implies that $$exists n_0inBbb{N}, forall n > n_0, frac 12*frac {1}{n^2}le u_nle frac 32*frac {1}{n^2}$$ Thus if $sum frac {1}{n^2}$ converges then so does $sum u_n$
– TheD0ubleT
Nov 28 at 8:16
Thanks for the answer. I believe you, that the series you built is convergent, but is there a more rigorous way to show that it converges other than "it converges because it approximates one over n-squared?"
– Justin
Nov 28 at 4:38
Thanks for the answer. I believe you, that the series you built is convergent, but is there a more rigorous way to show that it converges other than "it converges because it approximates one over n-squared?"
– Justin
Nov 28 at 4:38
It is rigorous, because $u_n sim frac {1}{n^2}$ implies that $$exists n_0inBbb{N}, forall n > n_0, frac 12*frac {1}{n^2}le u_nle frac 32*frac {1}{n^2}$$ Thus if $sum frac {1}{n^2}$ converges then so does $sum u_n$
– TheD0ubleT
Nov 28 at 8:16
It is rigorous, because $u_n sim frac {1}{n^2}$ implies that $$exists n_0inBbb{N}, forall n > n_0, frac 12*frac {1}{n^2}le u_nle frac 32*frac {1}{n^2}$$ Thus if $sum frac {1}{n^2}$ converges then so does $sum u_n$
– TheD0ubleT
Nov 28 at 8:16
add a comment |
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Hint: M-test. For all sufficiently large $n$, $$left|frac{1}{n(x-n)} right| leqslant frac{1}{n(n-b)}$$
– RRL
Nov 26 at 3:17
Thanks! I'll work getting an M-test to work with that inequality.
– Justin
Nov 26 at 20:06
I spoke with my Professor, and he agreed with you, although he mentioned that I'd need to use cases, as TheD0ubleT did.
– Justin
Nov 28 at 4:40