Difference between divergent series and series with no limit?
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Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.
sequences-and-series
asked Jun 18 '13 at 19:02
Aaron Booher
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Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.
sequences-and-series
asked Jun 18 '13 at 19:02
Aaron Booher
814
1
The limit could be $infty$ for example, in which case we still say it's divergent.
– Najib Idrissi
Jun 18 '13 at 19:04
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
– Ben Passer
Jun 18 '13 at 21:09
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.
sequences-and-series
asked Jun 18 '13 at 19:02
Aaron Booher
814
Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.
sequences-and-series
sequences-and-series
asked Jun 18 '13 at 19:02
Aaron Booher
814
asked Jun 18 '13 at 19:02
Aaron Booher
814
asked Jun 18 '13 at 19:02
Aaron Booher
814
asked Jun 18 '13 at 19:02
Aaron Booher
814
asked Jun 18 '13 at 19:02
Aaron Booher
814
814
1
The limit could be $infty$ for example, in which case we still say it's divergent.
– Najib Idrissi
Jun 18 '13 at 19:04
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
– Ben Passer
Jun 18 '13 at 21:09
add a comment |
1
The limit could be $infty$ for example, in which case we still say it's divergent.
– Najib Idrissi
Jun 18 '13 at 19:04
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
– Ben Passer
Jun 18 '13 at 21:09
1
1
The limit could be $infty$ for example, in which case we still say it's divergent.
– Najib Idrissi
Jun 18 '13 at 19:04
The limit could be $infty$ for example, in which case we still say it's divergent.
– Najib Idrissi
Jun 18 '13 at 19:04
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
– Ben Passer
Jun 18 '13 at 21:09
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
– Ben Passer
Jun 18 '13 at 21:09
add a comment |
2 Answers
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If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.
The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.
The partial sums obtain a finite limit if and only if the series is convergent.
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
– Math1000
May 27 '15 at 13:57
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In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:
A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).
Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.
Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.
edited Jan 7 '17 at 15:04
grg
1,0411812
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.
The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.
The partial sums obtain a finite limit if and only if the series is convergent.
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
– Math1000
May 27 '15 at 13:57
add a comment |
up vote
0
down vote
If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.
The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.
The partial sums obtain a finite limit if and only if the series is convergent.
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
– Math1000
May 27 '15 at 13:57
add a comment |
up vote
0
down vote
up vote
0
down vote
If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.
The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.
The partial sums obtain a finite limit if and only if the series is convergent.
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
If a series is convergent then it has bounded partial sums. Equivalently, if a series fails to have bounded partial sums then it is divergent.
The converse is false. If a series has bounded partial sums then it need not be convergent. A counter-example is $sum_{n ge 1}(-1)^n$ where the partial sums are $-1,0,-1,0,-1,ldots$.
The partial sums obtain a finite limit if and only if the series is convergent.
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
answered Jun 18 '13 at 19:20
Fly by Night
25.5k32977
25.5k32977
Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
– Math1000
May 27 '15 at 13:57
add a comment |
Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
– Math1000
May 27 '15 at 13:57
Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
– Math1000
May 27 '15 at 13:57
Boundedness of partial sums is a sufficient condition if the terms are nonnegative, though.
– Math1000
May 27 '15 at 13:57
add a comment |
up vote
0
down vote
In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:
A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).
Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.
Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.
edited Jan 7 '17 at 15:04
grg
1,0411812
add a comment |
up vote
0
down vote
In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:
A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).
Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.
Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.
edited Jan 7 '17 at 15:04
grg
1,0411812
add a comment |
up vote
0
down vote
up vote
0
down vote
In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:
A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).
Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.
Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.
edited Jan 7 '17 at 15:04
grg
1,0411812
In every calculus and real analysis book I've read, the two are the same thing.For example, in Stewart's calculus:
A sequence ${a_n}$ has the limit $L$ and we write $$lim_{ntoinfty}a_n=Lquadtext{or}quad a_nto Ltext{ as }ntoinfty$$ if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large. If $lim_{ntoinfty}$ exists, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).
Given a series $sum_{n=1}^infty a_n=a_1+a_2+a_3+cdots$, let $s_n$ denote its $n$th partial sum: $$s_n=sum_{i=1}^na_i=a_1+a_2+cdots+a_n$$ If the sequence ${s_n}$ is convergent and $lim_{ntoinfty}s_n=s$ exists as a real number, then the series $Sigma a_n$ is called convergent and we write $$a_1+a_2+cdots+a_n+cdots=squadtext{or}quadsum_{n=1}^infty a_n=s$$ The number $s$ is called the sum of the series. If the sequence ${s_n}$ is divergent, then the series is called divergent.
Either a series converges, or it diverges. Converge means it has a limit. Diverge means it doesn't.
edited Jan 7 '17 at 15:04
grg
1,0411812
edited Jan 7 '17 at 15:04
grg
1,0411812
edited Jan 7 '17 at 15:04
grg
1,0411812
edited Jan 7 '17 at 15:04
grg
1,0411812
1,0411812
answered Apr 25 '15 at 7:56
user223391
add a comment |
add a comment |
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The limit could be $infty$ for example, in which case we still say it's divergent.
– Najib Idrissi
Jun 18 '13 at 19:04
If you are reading this from a calculus book, make sure you are talking about the same objects having a limit. For example, a series $sum_{n = 1}^infty a_n$ converges if and only if the sequence of partial sums $s_k = sum_{n = 1}^k a_n$ has a limit as $k$ approaches $infty$. This is not the same as determining if the sequence $a_n$ has a limit as $n to infty$.
– Ben Passer
Jun 18 '13 at 21:09