Difference between a series and a sequence of partial sumsConvergence from partial sumsUnbounded Sequence...
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Difference between a series and a sequence of partial sums
Convergence from partial sumsUnbounded Sequence with Bounded Partial SumsSequence of Partial Sums for repeated decimalFind the sequence of partial sums for the series $a_n = (-1)^n$ Does this series converge?Help understanding the sequence of partial sums of a seriesSequence of partial sums converges to sum of seriesPartial Sums of equal seriesSequence of odd and even partial sums of alternating harmonic series.Limit of an infinite series as limit of sequence of partial sums.Then can the all terms of sequence of the partial sums of the series be strictly greater than zero?
$begingroup$
I understand what a series is.
I understand what a partial sum of a series is.
But what is a sequence of partial sums? As i understand it, it's the same thing that a series.
sequences-and-series
asked May 11 '17 at 20:09
HuguesHugues
156
$endgroup$
add a comment |
$begingroup$
I understand what a series is.
I understand what a partial sum of a series is.
But what is a sequence of partial sums? As i understand it, it's the same thing that a series.
sequences-and-series
asked May 11 '17 at 20:09
HuguesHugues
156
$endgroup$
1
$begingroup$
Correct. The convergence divergence of a series follows that of the sequence of partial sums. This is an example of economy ot thought ubiquitous in math. We don't need to rederive the same results again and can concentrate on the new ones instead.
$endgroup$
– Jyrki Lahtonen
May 11 '17 at 20:15
$begingroup$
A series is two sequences: i) a sequence $a_1,a_2, dots $ of summands, and ii) the sequence $a_1,a_1+a_2, dots $ of partial sums.
$endgroup$
– zhw.
May 11 '17 at 21:05
add a comment |
$begingroup$
I understand what a series is.
I understand what a partial sum of a series is.
But what is a sequence of partial sums? As i understand it, it's the same thing that a series.
sequences-and-series
asked May 11 '17 at 20:09
HuguesHugues
156
$endgroup$
I understand what a series is.
I understand what a partial sum of a series is.
But what is a sequence of partial sums? As i understand it, it's the same thing that a series.
sequences-and-series
sequences-and-series
asked May 11 '17 at 20:09
HuguesHugues
156
asked May 11 '17 at 20:09
HuguesHugues
156
asked May 11 '17 at 20:09
HuguesHugues
156
asked May 11 '17 at 20:09
HuguesHugues
156
asked May 11 '17 at 20:09
HuguesHugues
156
156
1
$begingroup$
Correct. The convergence divergence of a series follows that of the sequence of partial sums. This is an example of economy ot thought ubiquitous in math. We don't need to rederive the same results again and can concentrate on the new ones instead.
$endgroup$
– Jyrki Lahtonen
May 11 '17 at 20:15
$begingroup$
A series is two sequences: i) a sequence $a_1,a_2, dots $ of summands, and ii) the sequence $a_1,a_1+a_2, dots $ of partial sums.
$endgroup$
– zhw.
May 11 '17 at 21:05
add a comment |
1
$begingroup$
Correct. The convergence divergence of a series follows that of the sequence of partial sums. This is an example of economy ot thought ubiquitous in math. We don't need to rederive the same results again and can concentrate on the new ones instead.
$endgroup$
– Jyrki Lahtonen
May 11 '17 at 20:15
$begingroup$
A series is two sequences: i) a sequence $a_1,a_2, dots $ of summands, and ii) the sequence $a_1,a_1+a_2, dots $ of partial sums.
$endgroup$
– zhw.
May 11 '17 at 21:05
1
1
$begingroup$
Correct. The convergence divergence of a series follows that of the sequence of partial sums. This is an example of economy ot thought ubiquitous in math. We don't need to rederive the same results again and can concentrate on the new ones instead.
$endgroup$
– Jyrki Lahtonen
May 11 '17 at 20:15
$begingroup$
Correct. The convergence divergence of a series follows that of the sequence of partial sums. This is an example of economy ot thought ubiquitous in math. We don't need to rederive the same results again and can concentrate on the new ones instead.
$endgroup$
– Jyrki Lahtonen
May 11 '17 at 20:15
$begingroup$
A series is two sequences: i) a sequence $a_1,a_2, dots $ of summands, and ii) the sequence $a_1,a_1+a_2, dots $ of partial sums.
$endgroup$
– zhw.
May 11 '17 at 21:05
$begingroup$
A series is two sequences: i) a sequence $a_1,a_2, dots $ of summands, and ii) the sequence $a_1,a_1+a_2, dots $ of partial sums.
$endgroup$
– zhw.
May 11 '17 at 21:05
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Consider summming up the values $x^n$ for a fixed $x in [-1, 1]$. So you have
1) Series
The elements are $x$, $x^2$, $x^3$, $x^4$, ...
2) Sum of the series:
$$S = sum_{n=0}^infty x^n = frac 1{1-x}$$
3) Partial sum up to index $n$
$$S_n = sum_{k=0}^n x^k = frac{x^{n+1}-1}{x-1}$$
4) Sequence of partial sums of the series
Is the sequence $S_0, S_1, S_2, dots$. That is, is the sequence
$1, x, 1+x+x^2, dots$ which thanks to the formula above can be written as $1, x, frac{x^3-1}{x-1}, dots$
Finally, note that the sum of the series is (usually, but not always) defined as the limit of the sequence of the partial sums. That is, we define the sum of the series $S$ to be
$$S = lim_{ntoinfty} S_n = lim_{ntoinfty} frac{x^{n+1}-1}{x-1} = frac 1{1-x}$$
which is indeed the expression I wrote at point 2).
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
$endgroup$
add a comment |
$begingroup$
The expression: $1+2+3+4+5+cdots$ is a series.
The sequence of partial sums of that series is: $1, 3, 6, 10, 15, ldots$.
Thus, a sequence of partial sums is related to a series. If the sequence of partial sums converges, as a sequence, then the corresponding series is said to be convergent as well, and to equal that convergent value.
Remember: a series is a sum; a sequence is a list.
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
$endgroup$
$begingroup$
A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum
$endgroup$
– Hugues
May 11 '17 at 20:19
$begingroup$
I'm not sure I understand your question.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just mean that a series is written with plus signs, and a sequence is written with commas.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing
$endgroup$
– Hugues
May 11 '17 at 20:27
$begingroup$
Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:29
|
show 6 more comments
$begingroup$
A series $sum u_n $ is defined by the couple $(u _n,S_n) $ where $(S_n )_{ngeq n_0}$ is the sequence of partial sums given by
$$S_n=u_{n_0}+u_{n_0+1}+...u_n .$$
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Consider summming up the values $x^n$ for a fixed $x in [-1, 1]$. So you have
1) Series
The elements are $x$, $x^2$, $x^3$, $x^4$, ...
2) Sum of the series:
$$S = sum_{n=0}^infty x^n = frac 1{1-x}$$
3) Partial sum up to index $n$
$$S_n = sum_{k=0}^n x^k = frac{x^{n+1}-1}{x-1}$$
4) Sequence of partial sums of the series
Is the sequence $S_0, S_1, S_2, dots$. That is, is the sequence
$1, x, 1+x+x^2, dots$ which thanks to the formula above can be written as $1, x, frac{x^3-1}{x-1}, dots$
Finally, note that the sum of the series is (usually, but not always) defined as the limit of the sequence of the partial sums. That is, we define the sum of the series $S$ to be
$$S = lim_{ntoinfty} S_n = lim_{ntoinfty} frac{x^{n+1}-1}{x-1} = frac 1{1-x}$$
which is indeed the expression I wrote at point 2).
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
$endgroup$
add a comment |
$begingroup$
Consider summming up the values $x^n$ for a fixed $x in [-1, 1]$. So you have
1) Series
The elements are $x$, $x^2$, $x^3$, $x^4$, ...
2) Sum of the series:
$$S = sum_{n=0}^infty x^n = frac 1{1-x}$$
3) Partial sum up to index $n$
$$S_n = sum_{k=0}^n x^k = frac{x^{n+1}-1}{x-1}$$
4) Sequence of partial sums of the series
Is the sequence $S_0, S_1, S_2, dots$. That is, is the sequence
$1, x, 1+x+x^2, dots$ which thanks to the formula above can be written as $1, x, frac{x^3-1}{x-1}, dots$
Finally, note that the sum of the series is (usually, but not always) defined as the limit of the sequence of the partial sums. That is, we define the sum of the series $S$ to be
$$S = lim_{ntoinfty} S_n = lim_{ntoinfty} frac{x^{n+1}-1}{x-1} = frac 1{1-x}$$
which is indeed the expression I wrote at point 2).
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
$endgroup$
add a comment |
$begingroup$
Consider summming up the values $x^n$ for a fixed $x in [-1, 1]$. So you have
1) Series
The elements are $x$, $x^2$, $x^3$, $x^4$, ...
2) Sum of the series:
$$S = sum_{n=0}^infty x^n = frac 1{1-x}$$
3) Partial sum up to index $n$
$$S_n = sum_{k=0}^n x^k = frac{x^{n+1}-1}{x-1}$$
4) Sequence of partial sums of the series
Is the sequence $S_0, S_1, S_2, dots$. That is, is the sequence
$1, x, 1+x+x^2, dots$ which thanks to the formula above can be written as $1, x, frac{x^3-1}{x-1}, dots$
Finally, note that the sum of the series is (usually, but not always) defined as the limit of the sequence of the partial sums. That is, we define the sum of the series $S$ to be
$$S = lim_{ntoinfty} S_n = lim_{ntoinfty} frac{x^{n+1}-1}{x-1} = frac 1{1-x}$$
which is indeed the expression I wrote at point 2).
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
$endgroup$
Consider summming up the values $x^n$ for a fixed $x in [-1, 1]$. So you have
1) Series
The elements are $x$, $x^2$, $x^3$, $x^4$, ...
2) Sum of the series:
$$S = sum_{n=0}^infty x^n = frac 1{1-x}$$
3) Partial sum up to index $n$
$$S_n = sum_{k=0}^n x^k = frac{x^{n+1}-1}{x-1}$$
4) Sequence of partial sums of the series
Is the sequence $S_0, S_1, S_2, dots$. That is, is the sequence
$1, x, 1+x+x^2, dots$ which thanks to the formula above can be written as $1, x, frac{x^3-1}{x-1}, dots$
Finally, note that the sum of the series is (usually, but not always) defined as the limit of the sequence of the partial sums. That is, we define the sum of the series $S$ to be
$$S = lim_{ntoinfty} S_n = lim_{ntoinfty} frac{x^{n+1}-1}{x-1} = frac 1{1-x}$$
which is indeed the expression I wrote at point 2).
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
answered Aug 7 '18 at 9:11
AntAnt
17.5k23074
17.5k23074
add a comment |
add a comment |
$begingroup$
The expression: $1+2+3+4+5+cdots$ is a series.
The sequence of partial sums of that series is: $1, 3, 6, 10, 15, ldots$.
Thus, a sequence of partial sums is related to a series. If the sequence of partial sums converges, as a sequence, then the corresponding series is said to be convergent as well, and to equal that convergent value.
Remember: a series is a sum; a sequence is a list.
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
$endgroup$
$begingroup$
A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum
$endgroup$
– Hugues
May 11 '17 at 20:19
$begingroup$
I'm not sure I understand your question.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just mean that a series is written with plus signs, and a sequence is written with commas.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing
$endgroup$
– Hugues
May 11 '17 at 20:27
$begingroup$
Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:29
|
show 6 more comments
$begingroup$
The expression: $1+2+3+4+5+cdots$ is a series.
The sequence of partial sums of that series is: $1, 3, 6, 10, 15, ldots$.
Thus, a sequence of partial sums is related to a series. If the sequence of partial sums converges, as a sequence, then the corresponding series is said to be convergent as well, and to equal that convergent value.
Remember: a series is a sum; a sequence is a list.
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
$endgroup$
$begingroup$
A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum
$endgroup$
– Hugues
May 11 '17 at 20:19
$begingroup$
I'm not sure I understand your question.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just mean that a series is written with plus signs, and a sequence is written with commas.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing
$endgroup$
– Hugues
May 11 '17 at 20:27
$begingroup$
Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:29
|
show 6 more comments
$begingroup$
The expression: $1+2+3+4+5+cdots$ is a series.
The sequence of partial sums of that series is: $1, 3, 6, 10, 15, ldots$.
Thus, a sequence of partial sums is related to a series. If the sequence of partial sums converges, as a sequence, then the corresponding series is said to be convergent as well, and to equal that convergent value.
Remember: a series is a sum; a sequence is a list.
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
$endgroup$
The expression: $1+2+3+4+5+cdots$ is a series.
The sequence of partial sums of that series is: $1, 3, 6, 10, 15, ldots$.
Thus, a sequence of partial sums is related to a series. If the sequence of partial sums converges, as a sequence, then the corresponding series is said to be convergent as well, and to equal that convergent value.
Remember: a series is a sum; a sequence is a list.
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
answered May 11 '17 at 20:14
G Tony JacobsG Tony Jacobs
25.9k43686
25.9k43686
$begingroup$
A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum
$endgroup$
– Hugues
May 11 '17 at 20:19
$begingroup$
I'm not sure I understand your question.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just mean that a series is written with plus signs, and a sequence is written with commas.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing
$endgroup$
– Hugues
May 11 '17 at 20:27
$begingroup$
Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:29
|
show 6 more comments
$begingroup$
A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum
$endgroup$
– Hugues
May 11 '17 at 20:19
$begingroup$
I'm not sure I understand your question.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just mean that a series is written with plus signs, and a sequence is written with commas.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing
$endgroup$
– Hugues
May 11 '17 at 20:27
$begingroup$
Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:29
$begingroup$
A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum
$endgroup$
– Hugues
May 11 '17 at 20:19
$begingroup$
A series is a series is a sequence and a sequence of partial sums is a sequence too. So why one is a sum and the other a list? 6 = 1 + 2 + 3, it's also a sum
$endgroup$
– Hugues
May 11 '17 at 20:19
$begingroup$
I'm not sure I understand your question.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I'm not sure I understand your question.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just mean that a series is written with plus signs, and a sequence is written with commas.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just mean that a series is written with plus signs, and a sequence is written with commas.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:23
$begingroup$
I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing
$endgroup$
– Hugues
May 11 '17 at 20:27
$begingroup$
I just don't understand why is there a difference between a sequence of partial sums and a series. If we take u_(n+1) = u_n +1 (with u_0 = 0), the sequence of it's partial sum will be 1, 3, 6, 15... The series will be exactly the same thing
$endgroup$
– Hugues
May 11 '17 at 20:27
$begingroup$
Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:29
$begingroup$
Not really. The fourth term of the series is 4. The fourth term of the sequence is 10. It's helpful to see them as related, but semantically, we talk about them in different ways.
$endgroup$
– G Tony Jacobs
May 11 '17 at 20:29
|
show 6 more comments
$begingroup$
A series $sum u_n $ is defined by the couple $(u _n,S_n) $ where $(S_n )_{ngeq n_0}$ is the sequence of partial sums given by
$$S_n=u_{n_0}+u_{n_0+1}+...u_n .$$
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
add a comment |
$begingroup$
A series $sum u_n $ is defined by the couple $(u _n,S_n) $ where $(S_n )_{ngeq n_0}$ is the sequence of partial sums given by
$$S_n=u_{n_0}+u_{n_0+1}+...u_n .$$
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
add a comment |
$begingroup$
A series $sum u_n $ is defined by the couple $(u _n,S_n) $ where $(S_n )_{ngeq n_0}$ is the sequence of partial sums given by
$$S_n=u_{n_0}+u_{n_0+1}+...u_n .$$
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
$endgroup$
A series $sum u_n $ is defined by the couple $(u _n,S_n) $ where $(S_n )_{ngeq n_0}$ is the sequence of partial sums given by
$$S_n=u_{n_0}+u_{n_0+1}+...u_n .$$
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
answered May 11 '17 at 20:44
hamam_Abdallahhamam_Abdallah
38.2k21634
38.2k21634
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$begingroup$
Correct. The convergence divergence of a series follows that of the sequence of partial sums. This is an example of economy ot thought ubiquitous in math. We don't need to rederive the same results again and can concentrate on the new ones instead.
$endgroup$
– Jyrki Lahtonen
May 11 '17 at 20:15
$begingroup$
A series is two sequences: i) a sequence $a_1,a_2, dots $ of summands, and ii) the sequence $a_1,a_1+a_2, dots $ of partial sums.
$endgroup$
– zhw.
May 11 '17 at 21:05