Easy question on infinite series.
up vote
-1
down vote
favorite
$$sum_{k=0}^{infty} 1^k-1^{2k}.$$
On the one hand partial sums of this series are equal to 0, so our infinite series must converge to 0.
On the other hand
$$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}=sum_{k=0}^{infty} 1^{2k+1}$$
Obviously this series is not converge to 0. And i cant understand what rules i'm breaking?
real-analysis
add a comment |
up vote
-1
down vote
favorite
$$sum_{k=0}^{infty} 1^k-1^{2k}.$$
On the one hand partial sums of this series are equal to 0, so our infinite series must converge to 0.
On the other hand
$$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}=sum_{k=0}^{infty} 1^{2k+1}$$
Obviously this series is not converge to 0. And i cant understand what rules i'm breaking?
real-analysis
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
$$sum_{k=0}^{infty} 1^k-1^{2k}.$$
On the one hand partial sums of this series are equal to 0, so our infinite series must converge to 0.
On the other hand
$$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}=sum_{k=0}^{infty} 1^{2k+1}$$
Obviously this series is not converge to 0. And i cant understand what rules i'm breaking?
real-analysis
$$sum_{k=0}^{infty} 1^k-1^{2k}.$$
On the one hand partial sums of this series are equal to 0, so our infinite series must converge to 0.
On the other hand
$$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}=sum_{k=0}^{infty} 1^{2k+1}$$
Obviously this series is not converge to 0. And i cant understand what rules i'm breaking?
real-analysis
real-analysis
edited Nov 21 at 8:08
Matti P.
1,681413
1,681413
asked Nov 21 at 8:03
A.Kazakov
63
63
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}neqsum_{k=0}^{infty} 1^{2k+1}$
I do not understand why you think the last expression is equal to your sum. Are you able to explain why you believe that is the case?
Because the first addend is sum of all degrees of 1 and the second addend is the sum of all even degrees of 1, so their difference is the sum of all odd degrees of 1.
– A.Kazakov
Nov 21 at 8:30
Oh I see now. I think this is one of those cases where you can't change the order of your summation. So your original sum was obviously 0 (as you stated), I'm not sure that is correct to split it into two sums. I'll try and find an example of when changing the order of summation produces rubbish
– KnowsNothing
Nov 21 at 8:37
I suppose that Yves Daoust's answer to this question math.stackexchange.com/questions/657241/… is a good example of this kind of stuff.
– KnowsNothing
Nov 21 at 8:41
Thank you very much. I got it.
– A.Kazakov
Nov 21 at 18:57
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}neqsum_{k=0}^{infty} 1^{2k+1}$
I do not understand why you think the last expression is equal to your sum. Are you able to explain why you believe that is the case?
Because the first addend is sum of all degrees of 1 and the second addend is the sum of all even degrees of 1, so their difference is the sum of all odd degrees of 1.
– A.Kazakov
Nov 21 at 8:30
Oh I see now. I think this is one of those cases where you can't change the order of your summation. So your original sum was obviously 0 (as you stated), I'm not sure that is correct to split it into two sums. I'll try and find an example of when changing the order of summation produces rubbish
– KnowsNothing
Nov 21 at 8:37
I suppose that Yves Daoust's answer to this question math.stackexchange.com/questions/657241/… is a good example of this kind of stuff.
– KnowsNothing
Nov 21 at 8:41
Thank you very much. I got it.
– A.Kazakov
Nov 21 at 18:57
add a comment |
up vote
0
down vote
accepted
$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}neqsum_{k=0}^{infty} 1^{2k+1}$
I do not understand why you think the last expression is equal to your sum. Are you able to explain why you believe that is the case?
Because the first addend is sum of all degrees of 1 and the second addend is the sum of all even degrees of 1, so their difference is the sum of all odd degrees of 1.
– A.Kazakov
Nov 21 at 8:30
Oh I see now. I think this is one of those cases where you can't change the order of your summation. So your original sum was obviously 0 (as you stated), I'm not sure that is correct to split it into two sums. I'll try and find an example of when changing the order of summation produces rubbish
– KnowsNothing
Nov 21 at 8:37
I suppose that Yves Daoust's answer to this question math.stackexchange.com/questions/657241/… is a good example of this kind of stuff.
– KnowsNothing
Nov 21 at 8:41
Thank you very much. I got it.
– A.Kazakov
Nov 21 at 18:57
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}neqsum_{k=0}^{infty} 1^{2k+1}$
I do not understand why you think the last expression is equal to your sum. Are you able to explain why you believe that is the case?
$sum_{k=0}^{infty} 1^k-1^{2k}= sum_{k=0}^{infty} 1^k-sum_{k=0}^{infty} 1^{2k}neqsum_{k=0}^{infty} 1^{2k+1}$
I do not understand why you think the last expression is equal to your sum. Are you able to explain why you believe that is the case?
answered Nov 21 at 8:16
KnowsNothing
355
355
Because the first addend is sum of all degrees of 1 and the second addend is the sum of all even degrees of 1, so their difference is the sum of all odd degrees of 1.
– A.Kazakov
Nov 21 at 8:30
Oh I see now. I think this is one of those cases where you can't change the order of your summation. So your original sum was obviously 0 (as you stated), I'm not sure that is correct to split it into two sums. I'll try and find an example of when changing the order of summation produces rubbish
– KnowsNothing
Nov 21 at 8:37
I suppose that Yves Daoust's answer to this question math.stackexchange.com/questions/657241/… is a good example of this kind of stuff.
– KnowsNothing
Nov 21 at 8:41
Thank you very much. I got it.
– A.Kazakov
Nov 21 at 18:57
add a comment |
Because the first addend is sum of all degrees of 1 and the second addend is the sum of all even degrees of 1, so their difference is the sum of all odd degrees of 1.
– A.Kazakov
Nov 21 at 8:30
Oh I see now. I think this is one of those cases where you can't change the order of your summation. So your original sum was obviously 0 (as you stated), I'm not sure that is correct to split it into two sums. I'll try and find an example of when changing the order of summation produces rubbish
– KnowsNothing
Nov 21 at 8:37
I suppose that Yves Daoust's answer to this question math.stackexchange.com/questions/657241/… is a good example of this kind of stuff.
– KnowsNothing
Nov 21 at 8:41
Thank you very much. I got it.
– A.Kazakov
Nov 21 at 18:57
Because the first addend is sum of all degrees of 1 and the second addend is the sum of all even degrees of 1, so their difference is the sum of all odd degrees of 1.
– A.Kazakov
Nov 21 at 8:30
Because the first addend is sum of all degrees of 1 and the second addend is the sum of all even degrees of 1, so their difference is the sum of all odd degrees of 1.
– A.Kazakov
Nov 21 at 8:30
Oh I see now. I think this is one of those cases where you can't change the order of your summation. So your original sum was obviously 0 (as you stated), I'm not sure that is correct to split it into two sums. I'll try and find an example of when changing the order of summation produces rubbish
– KnowsNothing
Nov 21 at 8:37
Oh I see now. I think this is one of those cases where you can't change the order of your summation. So your original sum was obviously 0 (as you stated), I'm not sure that is correct to split it into two sums. I'll try and find an example of when changing the order of summation produces rubbish
– KnowsNothing
Nov 21 at 8:37
I suppose that Yves Daoust's answer to this question math.stackexchange.com/questions/657241/… is a good example of this kind of stuff.
– KnowsNothing
Nov 21 at 8:41
I suppose that Yves Daoust's answer to this question math.stackexchange.com/questions/657241/… is a good example of this kind of stuff.
– KnowsNothing
Nov 21 at 8:41
Thank you very much. I got it.
– A.Kazakov
Nov 21 at 18:57
Thank you very much. I got it.
– A.Kazakov
Nov 21 at 18:57
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%2f3007410%2feasy-question-on-infinite-series%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