Is there such a series?
0
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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 $x$?
sequences-and-series analysis
edited Jan 20 at 19:48
J.G.
27.6k22843
asked Jan 20 at 13:30
Low Zhen HengLow Zhen Heng
11
$endgroup$
add a comment |
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 $x$?
sequences-and-series analysis
edited Jan 20 at 19:48
J.G.
27.6k22843
asked Jan 20 at 13:30
Low Zhen HengLow Zhen Heng
11
$endgroup$
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
add a comment |
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 $x$?
sequences-and-series analysis
edited Jan 20 at 19:48
J.G.
27.6k22843
asked Jan 20 at 13:30
Low Zhen HengLow Zhen Heng
11
$endgroup$
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
sequences-and-series analysis
edited Jan 20 at 19:48
J.G.
27.6k22843
asked Jan 20 at 13:30
Low Zhen HengLow Zhen Heng
11
edited Jan 20 at 19:48
J.G.
27.6k22843
asked Jan 20 at 13:30
Low Zhen HengLow Zhen Heng
11
edited Jan 20 at 19:48
J.G.
27.6k22843
edited Jan 20 at 19:48
J.G.
27.6k22843
edited Jan 20 at 19:48
J.G.
27.6k22843
27.6k22843
asked Jan 20 at 13:30
Low Zhen HengLow Zhen Heng
11
asked Jan 20 at 13:30
Low Zhen HengLow Zhen Heng
11
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
add a comment |
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
add a comment |
0
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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