Is there such a series?












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Is there a series of the form $$sum_{n=0}^{infty}a_nx^n$$ such that it is convergent on $$-1le x le 1$$ and divergent on all other real number $x$?










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  • 2




    $begingroup$
    What happens if $a_n=1/n^2$?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 19:39






  • 2




    $begingroup$
    @SangchuiLee Or something like that, as long as $a_0$ is finite.
    $endgroup$
    – J.G.
    Jan 20 at 19:48










  • $begingroup$
    @SangchulLee but isn't $frac{1}{n^2}$ similar to the harmonic series which is an infinite series?
    $endgroup$
    – Low Zhen Heng
    Jan 20 at 21:37










  • $begingroup$
    @LowZhenHeng Are you aware of the $p$-series?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 22:08
















0












$begingroup$


Is there a series of the form $$sum_{n=0}^{infty}a_nx^n$$ such that it is convergent on $$-1le x le 1$$ and divergent on all other real number $x$?










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    What happens if $a_n=1/n^2$?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 19:39






  • 2




    $begingroup$
    @SangchuiLee Or something like that, as long as $a_0$ is finite.
    $endgroup$
    – J.G.
    Jan 20 at 19:48










  • $begingroup$
    @SangchulLee but isn't $frac{1}{n^2}$ similar to the harmonic series which is an infinite series?
    $endgroup$
    – Low Zhen Heng
    Jan 20 at 21:37










  • $begingroup$
    @LowZhenHeng Are you aware of the $p$-series?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 22:08














0












0








0


1



$begingroup$


Is there a series of the form $$sum_{n=0}^{infty}a_nx^n$$ such that it is convergent on $$-1le x le 1$$ and divergent on all other real number $x$?










share|cite|improve this question











$endgroup$




Is there a series of the form $$sum_{n=0}^{infty}a_nx^n$$ such that it is convergent on $$-1le x le 1$$ and divergent on all other real number $x$?







sequences-and-series analysis






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 20 at 19:48









J.G.

27.6k22843




27.6k22843










asked Jan 20 at 13:30









Low Zhen HengLow Zhen Heng

11




11








  • 2




    $begingroup$
    What happens if $a_n=1/n^2$?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 19:39






  • 2




    $begingroup$
    @SangchuiLee Or something like that, as long as $a_0$ is finite.
    $endgroup$
    – J.G.
    Jan 20 at 19:48










  • $begingroup$
    @SangchulLee but isn't $frac{1}{n^2}$ similar to the harmonic series which is an infinite series?
    $endgroup$
    – Low Zhen Heng
    Jan 20 at 21:37










  • $begingroup$
    @LowZhenHeng Are you aware of the $p$-series?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 22:08














  • 2




    $begingroup$
    What happens if $a_n=1/n^2$?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 19:39






  • 2




    $begingroup$
    @SangchuiLee Or something like that, as long as $a_0$ is finite.
    $endgroup$
    – J.G.
    Jan 20 at 19:48










  • $begingroup$
    @SangchulLee but isn't $frac{1}{n^2}$ similar to the harmonic series which is an infinite series?
    $endgroup$
    – Low Zhen Heng
    Jan 20 at 21:37










  • $begingroup$
    @LowZhenHeng Are you aware of the $p$-series?
    $endgroup$
    – Sangchul Lee
    Jan 20 at 22:08








2




2




$begingroup$
What happens if $a_n=1/n^2$?
$endgroup$
– Sangchul Lee
Jan 20 at 19:39




$begingroup$
What happens if $a_n=1/n^2$?
$endgroup$
– Sangchul Lee
Jan 20 at 19:39




2




2




$begingroup$
@SangchuiLee Or something like that, as long as $a_0$ is finite.
$endgroup$
– J.G.
Jan 20 at 19:48




$begingroup$
@SangchuiLee Or something like that, as long as $a_0$ is finite.
$endgroup$
– J.G.
Jan 20 at 19:48












$begingroup$
@SangchulLee but isn't $frac{1}{n^2}$ similar to the harmonic series which is an infinite series?
$endgroup$
– Low Zhen Heng
Jan 20 at 21:37




$begingroup$
@SangchulLee but isn't $frac{1}{n^2}$ similar to the harmonic series which is an infinite series?
$endgroup$
– Low Zhen Heng
Jan 20 at 21:37












$begingroup$
@LowZhenHeng Are you aware of the $p$-series?
$endgroup$
– Sangchul Lee
Jan 20 at 22:08




$begingroup$
@LowZhenHeng Are you aware of the $p$-series?
$endgroup$
– Sangchul Lee
Jan 20 at 22:08










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