Example for this criterion?

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Here is a powerful criterion on sequences and series :




Let $(a_n)_nin mathbbN in mathbbR^+^mathbbN$ a decreasing sequence. If $A_n = sum limits_k=0^na_k$ converges then $a_n = o(frac1n)$.




I was looking for an interesting example which uses this criterion.



Thanks in advance !




real-analysis sequences-and-series limits asymptotics




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  • This just means the summands decay quickly enough.
    – Sean Roberson
    Jul 16 at 2:07










  • Isn't this like a way to rephrase the test for $p$-series?
    – stressed out
    Jul 16 at 2:19











  • @stressedout test for $p$-series ?
    – Maman
    Jul 16 at 2:22











  • Yeah. Read Chris's comment. In other words, what you have said probably can be linked to the test for $p$-series by considering $a_n$ equivalent to a rational function in $n$ or something.
    – stressed out
    Jul 16 at 2:26














up vote
0
down vote

favorite












Here is a powerful criterion on sequences and series :




Let $(a_n)_nin mathbbN in mathbbR^+^mathbbN$ a decreasing sequence. If $A_n = sum limits_k=0^na_k$ converges then $a_n = o(frac1n)$.




I was looking for an interesting example which uses this criterion.



Thanks in advance !




real-analysis sequences-and-series limits asymptotics




share|cite|improve this question





















  • This just means the summands decay quickly enough.
    – Sean Roberson
    Jul 16 at 2:07










  • Isn't this like a way to rephrase the test for $p$-series?
    – stressed out
    Jul 16 at 2:19











  • @stressedout test for $p$-series ?
    – Maman
    Jul 16 at 2:22











  • Yeah. Read Chris's comment. In other words, what you have said probably can be linked to the test for $p$-series by considering $a_n$ equivalent to a rational function in $n$ or something.
    – stressed out
    Jul 16 at 2:26












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Here is a powerful criterion on sequences and series :




Let $(a_n)_nin mathbbN in mathbbR^+^mathbbN$ a decreasing sequence. If $A_n = sum limits_k=0^na_k$ converges then $a_n = o(frac1n)$.




I was looking for an interesting example which uses this criterion.



Thanks in advance !




real-analysis sequences-and-series limits asymptotics




share|cite|improve this question













Here is a powerful criterion on sequences and series :




Let $(a_n)_nin mathbbN in mathbbR^+^mathbbN$ a decreasing sequence. If $A_n = sum limits_k=0^na_k$ converges then $a_n = o(frac1n)$.




I was looking for an interesting example which uses this criterion.



Thanks in advance !





real-analysis sequences-and-series limits asymptotics





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 16 at 2:11
























asked Jul 16 at 1:44









Maman

1,161720




1,161720











  • This just means the summands decay quickly enough.
    – Sean Roberson
    Jul 16 at 2:07










  • Isn't this like a way to rephrase the test for $p$-series?
    – stressed out
    Jul 16 at 2:19











  • @stressedout test for $p$-series ?
    – Maman
    Jul 16 at 2:22











  • Yeah. Read Chris's comment. In other words, what you have said probably can be linked to the test for $p$-series by considering $a_n$ equivalent to a rational function in $n$ or something.
    – stressed out
    Jul 16 at 2:26
















  • This just means the summands decay quickly enough.
    – Sean Roberson
    Jul 16 at 2:07










  • Isn't this like a way to rephrase the test for $p$-series?
    – stressed out
    Jul 16 at 2:19











  • @stressedout test for $p$-series ?
    – Maman
    Jul 16 at 2:22











  • Yeah. Read Chris's comment. In other words, what you have said probably can be linked to the test for $p$-series by considering $a_n$ equivalent to a rational function in $n$ or something.
    – stressed out
    Jul 16 at 2:26















This just means the summands decay quickly enough.
– Sean Roberson
Jul 16 at 2:07




This just means the summands decay quickly enough.
– Sean Roberson
Jul 16 at 2:07












Isn't this like a way to rephrase the test for $p$-series?
– stressed out
Jul 16 at 2:19





Isn't this like a way to rephrase the test for $p$-series?
– stressed out
Jul 16 at 2:19













@stressedout test for $p$-series ?
– Maman
Jul 16 at 2:22





@stressedout test for $p$-series ?
– Maman
Jul 16 at 2:22













Yeah. Read Chris's comment. In other words, what you have said probably can be linked to the test for $p$-series by considering $a_n$ equivalent to a rational function in $n$ or something.
– stressed out
Jul 16 at 2:26




Yeah. Read Chris's comment. In other words, what you have said probably can be linked to the test for $p$-series by considering $a_n$ equivalent to a rational function in $n$ or something.
– stressed out
Jul 16 at 2:26










1 Answer
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1
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Interestingly, while the harmonic series $sumfrac1n$ diverges (and, of course, $frac1nnot=mathcal o (frac1n)$), $sum(frac1n)^1+epsilon$ converges for any $epsilon gt0$.



So the condition you have given is in a sense optimal...






share|cite|improve this answer





















  • Let $s>0$ an integer, then we define the finite set $E_s=nin mathbbN / a_nge frac1s $ and denote by $K_s$ the size of $E_s$. Then apparently $K_s= o(s)$.
    – Maman
    Jul 16 at 2:33











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













Interestingly, while the harmonic series $sumfrac1n$ diverges (and, of course, $frac1nnot=mathcal o (frac1n)$), $sum(frac1n)^1+epsilon$ converges for any $epsilon gt0$.



So the condition you have given is in a sense optimal...






share|cite|improve this answer





















  • Let $s>0$ an integer, then we define the finite set $E_s=nin mathbbN / a_nge frac1s $ and denote by $K_s$ the size of $E_s$. Then apparently $K_s= o(s)$.
    – Maman
    Jul 16 at 2:33















up vote
1
down vote













Interestingly, while the harmonic series $sumfrac1n$ diverges (and, of course, $frac1nnot=mathcal o (frac1n)$), $sum(frac1n)^1+epsilon$ converges for any $epsilon gt0$.



So the condition you have given is in a sense optimal...






share|cite|improve this answer





















  • Let $s>0$ an integer, then we define the finite set $E_s=nin mathbbN / a_nge frac1s $ and denote by $K_s$ the size of $E_s$. Then apparently $K_s= o(s)$.
    – Maman
    Jul 16 at 2:33













up vote
1
down vote










up vote
1
down vote









Interestingly, while the harmonic series $sumfrac1n$ diverges (and, of course, $frac1nnot=mathcal o (frac1n)$), $sum(frac1n)^1+epsilon$ converges for any $epsilon gt0$.



So the condition you have given is in a sense optimal...






share|cite|improve this answer













Interestingly, while the harmonic series $sumfrac1n$ diverges (and, of course, $frac1nnot=mathcal o (frac1n)$), $sum(frac1n)^1+epsilon$ converges for any $epsilon gt0$.



So the condition you have given is in a sense optimal...







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 16 at 2:21









Chris Custer

5,4582622




5,4582622











  • Let $s>0$ an integer, then we define the finite set $E_s=nin mathbbN / a_nge frac1s $ and denote by $K_s$ the size of $E_s$. Then apparently $K_s= o(s)$.
    – Maman
    Jul 16 at 2:33

















  • Let $s>0$ an integer, then we define the finite set $E_s=nin mathbbN / a_nge frac1s $ and denote by $K_s$ the size of $E_s$. Then apparently $K_s= o(s)$.
    – Maman
    Jul 16 at 2:33
















Let $s>0$ an integer, then we define the finite set $E_s=nin mathbbN / a_nge frac1s $ and denote by $K_s$ the size of $E_s$. Then apparently $K_s= o(s)$.
– Maman
Jul 16 at 2:33





Let $s>0$ an integer, then we define the finite set $E_s=nin mathbbN / a_nge frac1s $ and denote by $K_s$ the size of $E_s$. Then apparently $K_s= o(s)$.
– Maman
Jul 16 at 2:33













 

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