If the partial limits of two sequences are equals then the sequences equals
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I didn't understand something about partial limits of sequences:
Say there are two series - $a_n , b_n$. Its unknown if they converge.
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
What about the other way around - if the series are equal then the partial limits are equal?
The first one seem wrong to me but I can't find an example to contradict it.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N : a_n=b_n$ ? Does it changes the answer for any of the claims?
Just started learning about limits last month,
Thank you in advance for your answer.
calculus sequences-and-series limits
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
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add a comment |
$begingroup$
I didn't understand something about partial limits of sequences:
Say there are two series - $a_n , b_n$. Its unknown if they converge.
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
What about the other way around - if the series are equal then the partial limits are equal?
The first one seem wrong to me but I can't find an example to contradict it.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N : a_n=b_n$ ? Does it changes the answer for any of the claims?
Just started learning about limits last month,
Thank you in advance for your answer.
calculus sequences-and-series limits
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
$endgroup$
$begingroup$
Please edit the title to something more descriptive :)
$endgroup$
– Shaun
Nov 26 '18 at 12:59
add a comment |
$begingroup$
I didn't understand something about partial limits of sequences:
Say there are two series - $a_n , b_n$. Its unknown if they converge.
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
What about the other way around - if the series are equal then the partial limits are equal?
The first one seem wrong to me but I can't find an example to contradict it.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N : a_n=b_n$ ? Does it changes the answer for any of the claims?
Just started learning about limits last month,
Thank you in advance for your answer.
calculus sequences-and-series limits
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
$endgroup$
I didn't understand something about partial limits of sequences:
Say there are two series - $a_n , b_n$. Its unknown if they converge.
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
What about the other way around - if the series are equal then the partial limits are equal?
The first one seem wrong to me but I can't find an example to contradict it.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N : a_n=b_n$ ? Does it changes the answer for any of the claims?
Just started learning about limits last month,
Thank you in advance for your answer.
calculus sequences-and-series limits
calculus sequences-and-series limits
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
edited Nov 26 '18 at 13:02
Dexa sh
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
asked Nov 26 '18 at 12:44
Dexa shDexa sh
32
32
$begingroup$
Please edit the title to something more descriptive :)
$endgroup$
– Shaun
Nov 26 '18 at 12:59
add a comment |
$begingroup$
Please edit the title to something more descriptive :)
$endgroup$
– Shaun
Nov 26 '18 at 12:59
$begingroup$
Please edit the title to something more descriptive :)
$endgroup$
– Shaun
Nov 26 '18 at 12:59
$begingroup$
Please edit the title to something more descriptive :)
$endgroup$
– Shaun
Nov 26 '18 at 12:59
add a comment |
1 Answer
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Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
No, not at all. For example, the limits (and therefore all partial limits) of $left(frac{1}{n}right)$ and $(0)$ coincide, but the sequences clearly are not equal.
What about the other way around - if the series are equal then the partial limits are equal?
Yes, this is obvious. The (partial) limits of $(a_n)$ are the partial limits of $(a_n)$.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N:a_n=b_n$ ? Does it changes the answer for any of the claims?
No: for the first, the counterexample above still works. For the second, yes, you can change the first finitely-many terms of a sequence without changing its limiting behaviour.
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
$endgroup$
$begingroup$
Thank you for your answer!
$endgroup$
– Dexa sh
Nov 26 '18 at 13:16
add a comment |
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$begingroup$
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
No, not at all. For example, the limits (and therefore all partial limits) of $left(frac{1}{n}right)$ and $(0)$ coincide, but the sequences clearly are not equal.
What about the other way around - if the series are equal then the partial limits are equal?
Yes, this is obvious. The (partial) limits of $(a_n)$ are the partial limits of $(a_n)$.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N:a_n=b_n$ ? Does it changes the answer for any of the claims?
No: for the first, the counterexample above still works. For the second, yes, you can change the first finitely-many terms of a sequence without changing its limiting behaviour.
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
$endgroup$
$begingroup$
Thank you for your answer!
$endgroup$
– Dexa sh
Nov 26 '18 at 13:16
add a comment |
$begingroup$
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
No, not at all. For example, the limits (and therefore all partial limits) of $left(frac{1}{n}right)$ and $(0)$ coincide, but the sequences clearly are not equal.
What about the other way around - if the series are equal then the partial limits are equal?
Yes, this is obvious. The (partial) limits of $(a_n)$ are the partial limits of $(a_n)$.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N:a_n=b_n$ ? Does it changes the answer for any of the claims?
No: for the first, the counterexample above still works. For the second, yes, you can change the first finitely-many terms of a sequence without changing its limiting behaviour.
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
$endgroup$
$begingroup$
Thank you for your answer!
$endgroup$
– Dexa sh
Nov 26 '18 at 13:16
add a comment |
$begingroup$
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
No, not at all. For example, the limits (and therefore all partial limits) of $left(frac{1}{n}right)$ and $(0)$ coincide, but the sequences clearly are not equal.
What about the other way around - if the series are equal then the partial limits are equal?
Yes, this is obvious. The (partial) limits of $(a_n)$ are the partial limits of $(a_n)$.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N:a_n=b_n$ ? Does it changes the answer for any of the claims?
No: for the first, the counterexample above still works. For the second, yes, you can change the first finitely-many terms of a sequence without changing its limiting behaviour.
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
$endgroup$
Is it right to say that if the partial limits of the two series are equal to each other then the series are equal ?
No, not at all. For example, the limits (and therefore all partial limits) of $left(frac{1}{n}right)$ and $(0)$ coincide, but the sequences clearly are not equal.
What about the other way around - if the series are equal then the partial limits are equal?
Yes, this is obvious. The (partial) limits of $(a_n)$ are the partial limits of $(a_n)$.
What about if they are only equal from a certain point? meaning - There is an $N$ that for every $n>N:a_n=b_n$ ? Does it changes the answer for any of the claims?
No: for the first, the counterexample above still works. For the second, yes, you can change the first finitely-many terms of a sequence without changing its limiting behaviour.
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
answered Nov 26 '18 at 13:11
user3482749user3482749
4,047818
4,047818
$begingroup$
Thank you for your answer!
$endgroup$
– Dexa sh
Nov 26 '18 at 13:16
add a comment |
$begingroup$
Thank you for your answer!
$endgroup$
– Dexa sh
Nov 26 '18 at 13:16
$begingroup$
Thank you for your answer!
$endgroup$
– Dexa sh
Nov 26 '18 at 13:16
$begingroup$
Thank you for your answer!
$endgroup$
– Dexa sh
Nov 26 '18 at 13:16
add a comment |
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$begingroup$
Please edit the title to something more descriptive :)
$endgroup$
– Shaun
Nov 26 '18 at 12:59