Power series that diverges whenver z is a root of unity
up vote
2
down vote
favorite
We know that $sum_{n=1}^infty frac{z^n}{n}$ converges if $|z|=1$ except when $z=1$.
From this it is simple to construct another series that converges whenever $z$ is a $m$th root of unity, where $m < M$ for some $M$, with $sum_{m=1}^M sum_{n=1}^infty frac{z^{mn}}{n}$.
Is it possible to construct a sum that diverges whenever $z$ is any root of unity? I suspect that $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{n}$ is not actually a convergent series and thus naively taking the limit of the above series will not work.
Update:
How about the series $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{nm}$? This one diverges for (arg z)/2pi irrational, but I expect that it should converge otherwise. If not, how about $m!$ in the denominator?
sequences-and-series power-series
asked Nov 23 at 7:07
SKK
655
add a comment |
up vote
2
down vote
favorite
We know that $sum_{n=1}^infty frac{z^n}{n}$ converges if $|z|=1$ except when $z=1$.
From this it is simple to construct another series that converges whenever $z$ is a $m$th root of unity, where $m < M$ for some $M$, with $sum_{m=1}^M sum_{n=1}^infty frac{z^{mn}}{n}$.
Is it possible to construct a sum that diverges whenever $z$ is any root of unity? I suspect that $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{n}$ is not actually a convergent series and thus naively taking the limit of the above series will not work.
Update:
How about the series $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{nm}$? This one diverges for (arg z)/2pi irrational, but I expect that it should converge otherwise. If not, how about $m!$ in the denominator?
sequences-and-series power-series
asked Nov 23 at 7:07
SKK
655
If a power series converges on all irrationals, then it should converge to all rationals inside the circle of convergence.
– Anupam
Nov 23 at 8:51
Sorry, poor explanation; a power series that converges on all |z|=1 with (arg z )/2pi irrational.
– SKK
Nov 23 at 15:41
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
We know that $sum_{n=1}^infty frac{z^n}{n}$ converges if $|z|=1$ except when $z=1$.
From this it is simple to construct another series that converges whenever $z$ is a $m$th root of unity, where $m < M$ for some $M$, with $sum_{m=1}^M sum_{n=1}^infty frac{z^{mn}}{n}$.
Is it possible to construct a sum that diverges whenever $z$ is any root of unity? I suspect that $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{n}$ is not actually a convergent series and thus naively taking the limit of the above series will not work.
Update:
How about the series $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{nm}$? This one diverges for (arg z)/2pi irrational, but I expect that it should converge otherwise. If not, how about $m!$ in the denominator?
sequences-and-series power-series
asked Nov 23 at 7:07
SKK
655
We know that $sum_{n=1}^infty frac{z^n}{n}$ converges if $|z|=1$ except when $z=1$.
From this it is simple to construct another series that converges whenever $z$ is a $m$th root of unity, where $m < M$ for some $M$, with $sum_{m=1}^M sum_{n=1}^infty frac{z^{mn}}{n}$.
Is it possible to construct a sum that diverges whenever $z$ is any root of unity? I suspect that $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{n}$ is not actually a convergent series and thus naively taking the limit of the above series will not work.
Update:
How about the series $sum_{m=1}^infty sum_{n=1}^infty frac{z^{mn}}{nm}$? This one diverges for (arg z)/2pi irrational, but I expect that it should converge otherwise. If not, how about $m!$ in the denominator?
sequences-and-series power-series
sequences-and-series power-series
asked Nov 23 at 7:07
SKK
655
asked Nov 23 at 7:07
SKK
655
edited Nov 23 at 16:03
asked Nov 23 at 7:07
SKK
655
asked Nov 23 at 7:07
SKK
655
asked Nov 23 at 7:07
SKK
655
655
If a power series converges on all irrationals, then it should converge to all rationals inside the circle of convergence.
– Anupam
Nov 23 at 8:51
Sorry, poor explanation; a power series that converges on all |z|=1 with (arg z )/2pi irrational.
– SKK
Nov 23 at 15:41
add a comment |
If a power series converges on all irrationals, then it should converge to all rationals inside the circle of convergence.
– Anupam
Nov 23 at 8:51
Sorry, poor explanation; a power series that converges on all |z|=1 with (arg z )/2pi irrational.
– SKK
Nov 23 at 15:41
If a power series converges on all irrationals, then it should converge to all rationals inside the circle of convergence.
– Anupam
Nov 23 at 8:51
If a power series converges on all irrationals, then it should converge to all rationals inside the circle of convergence.
– Anupam
Nov 23 at 8:51
Sorry, poor explanation; a power series that converges on all |z|=1 with (arg z )/2pi irrational.
– SKK
Nov 23 at 15:41
Sorry, poor explanation; a power series that converges on all |z|=1 with (arg z )/2pi irrational.
– SKK
Nov 23 at 15:41
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3010063%2fpower-series-that-diverges-whenver-z-is-a-root-of-unity%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
If a power series converges on all irrationals, then it should converge to all rationals inside the circle of convergence.
– Anupam
Nov 23 at 8:51
Sorry, poor explanation; a power series that converges on all |z|=1 with (arg z )/2pi irrational.
– SKK
Nov 23 at 15:41