Trying to Prove a generalization of Raabe's test
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I'm trying to prove (or disprove) a statement by Feld:
https://math.feld.cvut.cz/mt/txte/2/txe3ea2d.htm
It's a generalization of Raabe's test and it says that for a sequence $a_n$, for all n, if $a_n>0$ and $c_n$ is absolutely convergent and
$a_{n+1}/a_n = 1-A/n+c_n$
Then $sum_{n=1}^infty a_n$ converges iff $A>1$
I rewrite:
$n(1-a_{n+1}/a_n)=A-n c_n$
and take the limit on both sides as $n to infty$ and its proven by Raabe's simple test, as long as $lim_{n to infty} n c_n=0$.
But it has been shown (Series converges implies $lim{n a_n} = 0$) that $lim_{n to infty} n c_n$ is not zero unless $c_n$ is non-increasing, so I'm stuck. Did Feld forget to add that $c_n$ is non-increasing, or is he right as it stands?
real-analysis convergence
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
$endgroup$
add a comment |
$begingroup$
I'm trying to prove (or disprove) a statement by Feld:
https://math.feld.cvut.cz/mt/txte/2/txe3ea2d.htm
It's a generalization of Raabe's test and it says that for a sequence $a_n$, for all n, if $a_n>0$ and $c_n$ is absolutely convergent and
$a_{n+1}/a_n = 1-A/n+c_n$
Then $sum_{n=1}^infty a_n$ converges iff $A>1$
I rewrite:
$n(1-a_{n+1}/a_n)=A-n c_n$
and take the limit on both sides as $n to infty$ and its proven by Raabe's simple test, as long as $lim_{n to infty} n c_n=0$.
But it has been shown (Series converges implies $lim{n a_n} = 0$) that $lim_{n to infty} n c_n$ is not zero unless $c_n$ is non-increasing, so I'm stuck. Did Feld forget to add that $c_n$ is non-increasing, or is he right as it stands?
real-analysis convergence
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
$endgroup$
add a comment |
$begingroup$
I'm trying to prove (or disprove) a statement by Feld:
https://math.feld.cvut.cz/mt/txte/2/txe3ea2d.htm
It's a generalization of Raabe's test and it says that for a sequence $a_n$, for all n, if $a_n>0$ and $c_n$ is absolutely convergent and
$a_{n+1}/a_n = 1-A/n+c_n$
Then $sum_{n=1}^infty a_n$ converges iff $A>1$
I rewrite:
$n(1-a_{n+1}/a_n)=A-n c_n$
and take the limit on both sides as $n to infty$ and its proven by Raabe's simple test, as long as $lim_{n to infty} n c_n=0$.
But it has been shown (Series converges implies $lim{n a_n} = 0$) that $lim_{n to infty} n c_n$ is not zero unless $c_n$ is non-increasing, so I'm stuck. Did Feld forget to add that $c_n$ is non-increasing, or is he right as it stands?
real-analysis convergence
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
$endgroup$
I'm trying to prove (or disprove) a statement by Feld:
https://math.feld.cvut.cz/mt/txte/2/txe3ea2d.htm
It's a generalization of Raabe's test and it says that for a sequence $a_n$, for all n, if $a_n>0$ and $c_n$ is absolutely convergent and
$a_{n+1}/a_n = 1-A/n+c_n$
Then $sum_{n=1}^infty a_n$ converges iff $A>1$
I rewrite:
$n(1-a_{n+1}/a_n)=A-n c_n$
and take the limit on both sides as $n to infty$ and its proven by Raabe's simple test, as long as $lim_{n to infty} n c_n=0$.
But it has been shown (Series converges implies $lim{n a_n} = 0$) that $lim_{n to infty} n c_n$ is not zero unless $c_n$ is non-increasing, so I'm stuck. Did Feld forget to add that $c_n$ is non-increasing, or is he right as it stands?
real-analysis convergence
real-analysis convergence
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
edited Dec 9 '18 at 21:22
Paul R.
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
asked Dec 9 '18 at 19:29
Paul R.Paul R.
213
213
add a comment |
add a comment |
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