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Convergence test for partial sum whose elements all change as the index increases
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I have a sequence of length $n$, $b_i(n),, i=1,...,n$. Each $b_i(n)$ is a strictly positive rational number. I write $b_i(n)$ because indeed each element of the sequence is a function also of the length of the sequence, and as the sequence progresses all its elements change value.
If $S_n = sum_i=1^nb_i(n)$, I need to examine whether $lim_nto inftyS_n$ converges to a finite limit or not. Partial sums look like
$$S_n = sum_i=1^nb_i(n)$$
$$S_n+1 = sum_i=1^nb_i(n+1) + b_n+1(n+1)$$
I know that the limit of the partial sum will never be zero. I know that each element of the sequence tends to zero as $n$ progresses. But since I have infinite elements of the partial sum that go to zero, I am not sure how to determine whether the whole infinite series remains finite or not.
Can somebody suggest any method or point to relevant sources that examine this issue? I am not sure that the usual tests for convergence of a series/limit of a partial sum (like the ratio test etc) apply here.
sequences-and-series convergence
asked Aug 2 at 0:17
Alecos Papadopoulos
7,92811535
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I have a sequence of length $n$, $b_i(n),, i=1,...,n$. Each $b_i(n)$ is a strictly positive rational number. I write $b_i(n)$ because indeed each element of the sequence is a function also of the length of the sequence, and as the sequence progresses all its elements change value.
If $S_n = sum_i=1^nb_i(n)$, I need to examine whether $lim_nto inftyS_n$ converges to a finite limit or not. Partial sums look like
$$S_n = sum_i=1^nb_i(n)$$
$$S_n+1 = sum_i=1^nb_i(n+1) + b_n+1(n+1)$$
I know that the limit of the partial sum will never be zero. I know that each element of the sequence tends to zero as $n$ progresses. But since I have infinite elements of the partial sum that go to zero, I am not sure how to determine whether the whole infinite series remains finite or not.
Can somebody suggest any method or point to relevant sources that examine this issue? I am not sure that the usual tests for convergence of a series/limit of a partial sum (like the ratio test etc) apply here.
sequences-and-series convergence
asked Aug 2 at 0:17
Alecos Papadopoulos
7,92811535
1
I just want to make sure I understand. I'm think of this as an infinite lower triangular matrix, where $S_n$ is the sum of the $n$th row. When you say that each element of the sequence goes to $0,$ it means that each of the columns goes to $0$. Then you want to know whether that guarantees that the row sums go to a finite limit. Am I correct?
– saulspatz
Aug 2 at 2:17
This cannot be answered in general without more specifics or, even better, some example(s).each element of the sequence is a function also of the length of the sequenceThis is always the case for Riemann sums for example, see e.g. How do you calculate this limit $lim_ntoinftysum_k=1^n frackn^2+k^2$?, but this is by no means a general answer.
– dxiv
Aug 2 at 2:42
@saulspatz Yes, good image.
– Alecos Papadopoulos
Aug 2 at 5:34
add a comment |Â
up vote
0
down vote
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up vote
0
down vote
favorite
I have a sequence of length $n$, $b_i(n),, i=1,...,n$. Each $b_i(n)$ is a strictly positive rational number. I write $b_i(n)$ because indeed each element of the sequence is a function also of the length of the sequence, and as the sequence progresses all its elements change value.
If $S_n = sum_i=1^nb_i(n)$, I need to examine whether $lim_nto inftyS_n$ converges to a finite limit or not. Partial sums look like
$$S_n = sum_i=1^nb_i(n)$$
$$S_n+1 = sum_i=1^nb_i(n+1) + b_n+1(n+1)$$
I know that the limit of the partial sum will never be zero. I know that each element of the sequence tends to zero as $n$ progresses. But since I have infinite elements of the partial sum that go to zero, I am not sure how to determine whether the whole infinite series remains finite or not.
Can somebody suggest any method or point to relevant sources that examine this issue? I am not sure that the usual tests for convergence of a series/limit of a partial sum (like the ratio test etc) apply here.
sequences-and-series convergence
asked Aug 2 at 0:17
Alecos Papadopoulos
7,92811535
I have a sequence of length $n$, $b_i(n),, i=1,...,n$. Each $b_i(n)$ is a strictly positive rational number. I write $b_i(n)$ because indeed each element of the sequence is a function also of the length of the sequence, and as the sequence progresses all its elements change value.
If $S_n = sum_i=1^nb_i(n)$, I need to examine whether $lim_nto inftyS_n$ converges to a finite limit or not. Partial sums look like
$$S_n = sum_i=1^nb_i(n)$$
$$S_n+1 = sum_i=1^nb_i(n+1) + b_n+1(n+1)$$
I know that the limit of the partial sum will never be zero. I know that each element of the sequence tends to zero as $n$ progresses. But since I have infinite elements of the partial sum that go to zero, I am not sure how to determine whether the whole infinite series remains finite or not.
Can somebody suggest any method or point to relevant sources that examine this issue? I am not sure that the usual tests for convergence of a series/limit of a partial sum (like the ratio test etc) apply here.
sequences-and-series convergence
asked Aug 2 at 0:17
Alecos Papadopoulos
7,92811535
edited Aug 2 at 5:36
asked Aug 2 at 0:17
Alecos Papadopoulos
7,92811535
asked Aug 2 at 0:17
Alecos Papadopoulos
7,92811535
asked Aug 2 at 0:17
Alecos Papadopoulos
7,92811535
7,92811535
1
I just want to make sure I understand. I'm think of this as an infinite lower triangular matrix, where $S_n$ is the sum of the $n$th row. When you say that each element of the sequence goes to $0,$ it means that each of the columns goes to $0$. Then you want to know whether that guarantees that the row sums go to a finite limit. Am I correct?
– saulspatz
Aug 2 at 2:17
This cannot be answered in general without more specifics or, even better, some example(s).each element of the sequence is a function also of the length of the sequenceThis is always the case for Riemann sums for example, see e.g. How do you calculate this limit $lim_ntoinftysum_k=1^n frackn^2+k^2$?, but this is by no means a general answer.
– dxiv
Aug 2 at 2:42
@saulspatz Yes, good image.
– Alecos Papadopoulos
Aug 2 at 5:34
add a comment |Â
1
I just want to make sure I understand. I'm think of this as an infinite lower triangular matrix, where $S_n$ is the sum of the $n$th row. When you say that each element of the sequence goes to $0,$ it means that each of the columns goes to $0$. Then you want to know whether that guarantees that the row sums go to a finite limit. Am I correct?
– saulspatz
Aug 2 at 2:17
This cannot be answered in general without more specifics or, even better, some example(s).each element of the sequence is a function also of the length of the sequenceThis is always the case for Riemann sums for example, see e.g. How do you calculate this limit $lim_ntoinftysum_k=1^n frackn^2+k^2$?, but this is by no means a general answer.
– dxiv
Aug 2 at 2:42
@saulspatz Yes, good image.
– Alecos Papadopoulos
Aug 2 at 5:34
1
1
I just want to make sure I understand. I'm think of this as an infinite lower triangular matrix, where $S_n$ is the sum of the $n$th row. When you say that each element of the sequence goes to $0,$ it means that each of the columns goes to $0$. Then you want to know whether that guarantees that the row sums go to a finite limit. Am I correct?
– saulspatz
Aug 2 at 2:17
I just want to make sure I understand. I'm think of this as an infinite lower triangular matrix, where $S_n$ is the sum of the $n$th row. When you say that each element of the sequence goes to $0,$ it means that each of the columns goes to $0$. Then you want to know whether that guarantees that the row sums go to a finite limit. Am I correct?
– saulspatz
Aug 2 at 2:17
This cannot be answered in general without more specifics or, even better, some example(s).
each element of the sequence is a function also of the length of the sequence This is always the case for Riemann sums for example, see e.g. How do you calculate this limit $lim_ntoinftysum_k=1^n frackn^2+k^2$?, but this is by no means a general answer.– dxiv
Aug 2 at 2:42
This cannot be answered in general without more specifics or, even better, some example(s).
each element of the sequence is a function also of the length of the sequence This is always the case for Riemann sums for example, see e.g. How do you calculate this limit $lim_ntoinftysum_k=1^n frackn^2+k^2$?, but this is by no means a general answer.– dxiv
Aug 2 at 2:42
@saulspatz Yes, good image.
– Alecos Papadopoulos
Aug 2 at 5:34
@saulspatz Yes, good image.
– Alecos Papadopoulos
Aug 2 at 5:34
add a comment |Â
1 Answer
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If I've understood the question correctly, as indicated in my comment, then the answer is "no." For example set $$
b_i(n)=cases1over n,&$1leq ileq n, $if $n$ is odd\
2over n,&$1leq ileq n, $if $n$ is even
$$
so that $S_n=1$ if $n$ is odd, and $S_n=2$ is $n$ is even, and $S_n$ does not converge.
Or you could define $$b_i(n) =log nover n$$ to get $S_ntoinfty.$
EDIT
I just remembered that you want $b_i(n)$ to be rational. That makes no difference. Just take a good rational approximation to $log n.$
answered Aug 2 at 2:27
saulspatz
10.5k21324
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
If I've understood the question correctly, as indicated in my comment, then the answer is "no." For example set $$
b_i(n)=cases1over n,&$1leq ileq n, $if $n$ is odd\
2over n,&$1leq ileq n, $if $n$ is even
$$
so that $S_n=1$ if $n$ is odd, and $S_n=2$ is $n$ is even, and $S_n$ does not converge.
Or you could define $$b_i(n) =log nover n$$ to get $S_ntoinfty.$
EDIT
I just remembered that you want $b_i(n)$ to be rational. That makes no difference. Just take a good rational approximation to $log n.$
answered Aug 2 at 2:27
saulspatz
10.5k21324
add a comment |Â
up vote
1
down vote
If I've understood the question correctly, as indicated in my comment, then the answer is "no." For example set $$
b_i(n)=cases1over n,&$1leq ileq n, $if $n$ is odd\
2over n,&$1leq ileq n, $if $n$ is even
$$
so that $S_n=1$ if $n$ is odd, and $S_n=2$ is $n$ is even, and $S_n$ does not converge.
Or you could define $$b_i(n) =log nover n$$ to get $S_ntoinfty.$
EDIT
I just remembered that you want $b_i(n)$ to be rational. That makes no difference. Just take a good rational approximation to $log n.$
answered Aug 2 at 2:27
saulspatz
10.5k21324
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If I've understood the question correctly, as indicated in my comment, then the answer is "no." For example set $$
b_i(n)=cases1over n,&$1leq ileq n, $if $n$ is odd\
2over n,&$1leq ileq n, $if $n$ is even
$$
so that $S_n=1$ if $n$ is odd, and $S_n=2$ is $n$ is even, and $S_n$ does not converge.
Or you could define $$b_i(n) =log nover n$$ to get $S_ntoinfty.$
EDIT
I just remembered that you want $b_i(n)$ to be rational. That makes no difference. Just take a good rational approximation to $log n.$
answered Aug 2 at 2:27
saulspatz
10.5k21324
If I've understood the question correctly, as indicated in my comment, then the answer is "no." For example set $$
b_i(n)=cases1over n,&$1leq ileq n, $if $n$ is odd\
2over n,&$1leq ileq n, $if $n$ is even
$$
so that $S_n=1$ if $n$ is odd, and $S_n=2$ is $n$ is even, and $S_n$ does not converge.
Or you could define $$b_i(n) =log nover n$$ to get $S_ntoinfty.$
EDIT
I just remembered that you want $b_i(n)$ to be rational. That makes no difference. Just take a good rational approximation to $log n.$
answered Aug 2 at 2:27
saulspatz
10.5k21324
answered Aug 2 at 2:27
saulspatz
10.5k21324
answered Aug 2 at 2:27
saulspatz
10.5k21324
answered Aug 2 at 2:27
saulspatz
10.5k21324
10.5k21324
add a comment |Â
add a comment |Â
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1
I just want to make sure I understand. I'm think of this as an infinite lower triangular matrix, where $S_n$ is the sum of the $n$th row. When you say that each element of the sequence goes to $0,$ it means that each of the columns goes to $0$. Then you want to know whether that guarantees that the row sums go to a finite limit. Am I correct?
– saulspatz
Aug 2 at 2:17
This cannot be answered in general without more specifics or, even better, some example(s).
each element of the sequence is a function also of the length of the sequenceThis is always the case for Riemann sums for example, see e.g. How do you calculate this limit $lim_ntoinftysum_k=1^n frackn^2+k^2$?, but this is by no means a general answer.– dxiv
Aug 2 at 2:42
@saulspatz Yes, good image.
– Alecos Papadopoulos
Aug 2 at 5:34