If the partial limits of two sequences are equals then the sequences equals
up vote
0
down vote
favorite
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 at 12:44
Dexa sh
32
add a comment |
up vote
0
down vote
favorite
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 at 12:44
Dexa sh
32
Please edit the title to something more descriptive :)
– Shaun
Nov 26 at 12:59
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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 at 12:44
Dexa sh
32
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 at 12:44
Dexa sh
32
asked Nov 26 at 12:44
Dexa sh
32
edited Nov 26 at 13:02
asked Nov 26 at 12:44
Dexa sh
32
asked Nov 26 at 12:44
Dexa sh
32
asked Nov 26 at 12:44
Dexa sh
32
32
Please edit the title to something more descriptive :)
– Shaun
Nov 26 at 12:59
add a comment |
Please edit the title to something more descriptive :)
– Shaun
Nov 26 at 12:59
Please edit the title to something more descriptive :)
– Shaun
Nov 26 at 12:59
Please edit the title to something more descriptive :)
– Shaun
Nov 26 at 12:59
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
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 at 13:11
user3482749
2,096414
Thank you for your answer!
– Dexa sh
Nov 26 at 13:16
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014276%2fif-the-partial-limits-of-two-sequences-are-equals-then-the-sequences-equals%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
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 at 13:11
user3482749
2,096414
Thank you for your answer!
– Dexa sh
Nov 26 at 13:16
add a comment |
up vote
0
down vote
accepted
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 at 13:11
user3482749
2,096414
Thank you for your answer!
– Dexa sh
Nov 26 at 13:16
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
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 at 13:11
user3482749
2,096414
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 at 13:11
user3482749
2,096414
answered Nov 26 at 13:11
user3482749
2,096414
answered Nov 26 at 13:11
user3482749
2,096414
answered Nov 26 at 13:11
user3482749
2,096414
2,096414
Thank you for your answer!
– Dexa sh
Nov 26 at 13:16
add a comment |
Thank you for your answer!
– Dexa sh
Nov 26 at 13:16
Thank you for your answer!
– Dexa sh
Nov 26 at 13:16
Thank you for your answer!
– Dexa sh
Nov 26 at 13:16
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3014276%2fif-the-partial-limits-of-two-sequences-are-equals-then-the-sequences-equals%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Please edit the title to something more descriptive :)
– Shaun
Nov 26 at 12:59