Show the series $sumlimits^{infty}_{n=1}{frac{a_n}{n}}$ converges











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4
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Suppose the sequence ${a_n}_{n=1}^{infty}$ satisfies
$$midsumlimits^{n}_{k=1}{a_{k}}midleq Csqrt{n} spacespacespacespacespacespacespacespacespacespacespacespacespacespacespace n=1, 2, 3, cdots$$
where $C$ is a fixed (but arbitrary) number. Prove that the series
$$sumlimits^{infty}_{n=1}{frac{a_n}{n}}$$
converges.




My attempt: Suppose $b_n:= frac{1}{n}$; Abel's lemma on summation by parts gives
$$sumlimits^{k}_{n=1}{frac{a_n}{n}}=sumlimits^{k-1}_{n=1}{[sumlimits_{i=1}^{n}{a_i}cdot(b_n-b_{n+1})] + sumlimits_{i=1}^{k}{a_i}cdot b_k}$$



$$<sumlimits^{k-1}_{n=1}{midsumlimits_{i=1}^{n}{a_i}midcdot(b_n-b_{n+1}) +mid sumlimits_{i=1}^{k}{a_i}mid cdot b_k}$$



$$lesumlimits_{n=1}^{k}[Csqrt{n}cdot{(frac{1}{n}-frac{1}{n+1}})]+Csqrt{k}cdot frac{1}{k+1}$$



$$=sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}+frac{Csqrt{k}}{k+1}.$$



Since $krightarrowinfty$, therefore
$$frac{Csqrt{k}}{k+1}rightarrow 0.$$



Moreover, for the sigma notation, since
$$frac{Csqrt{n}}{n(n+1)}<frac{Csqrt{n}}{n^2}=frac{C}{n^frac{3}{2}}$$



Above is a $p$-series with $p=frac{3}{2}>1$, hence the series $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges. Even though $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges and $frac{Csqrt{k}}{k+1}$ approaches to $0$ for $krightarrowinfty$, but they do not imply the series $sumlimits^{infty}_{n=1}{frac{a_n}{n}}$ converges.



My proof seems not yet completed. How do I continue it?










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  • You switch between $k$ and $n$ in your solution, you should fix that.
    – Jakobian
    Nov 22 at 14:50










  • I had edited it. Thanks for pointing my mistake.
    – weilam06
    Nov 22 at 15:02















up vote
4
down vote

favorite













Suppose the sequence ${a_n}_{n=1}^{infty}$ satisfies
$$midsumlimits^{n}_{k=1}{a_{k}}midleq Csqrt{n} spacespacespacespacespacespacespacespacespacespacespacespacespacespacespace n=1, 2, 3, cdots$$
where $C$ is a fixed (but arbitrary) number. Prove that the series
$$sumlimits^{infty}_{n=1}{frac{a_n}{n}}$$
converges.




My attempt: Suppose $b_n:= frac{1}{n}$; Abel's lemma on summation by parts gives
$$sumlimits^{k}_{n=1}{frac{a_n}{n}}=sumlimits^{k-1}_{n=1}{[sumlimits_{i=1}^{n}{a_i}cdot(b_n-b_{n+1})] + sumlimits_{i=1}^{k}{a_i}cdot b_k}$$



$$<sumlimits^{k-1}_{n=1}{midsumlimits_{i=1}^{n}{a_i}midcdot(b_n-b_{n+1}) +mid sumlimits_{i=1}^{k}{a_i}mid cdot b_k}$$



$$lesumlimits_{n=1}^{k}[Csqrt{n}cdot{(frac{1}{n}-frac{1}{n+1}})]+Csqrt{k}cdot frac{1}{k+1}$$



$$=sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}+frac{Csqrt{k}}{k+1}.$$



Since $krightarrowinfty$, therefore
$$frac{Csqrt{k}}{k+1}rightarrow 0.$$



Moreover, for the sigma notation, since
$$frac{Csqrt{n}}{n(n+1)}<frac{Csqrt{n}}{n^2}=frac{C}{n^frac{3}{2}}$$



Above is a $p$-series with $p=frac{3}{2}>1$, hence the series $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges. Even though $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges and $frac{Csqrt{k}}{k+1}$ approaches to $0$ for $krightarrowinfty$, but they do not imply the series $sumlimits^{infty}_{n=1}{frac{a_n}{n}}$ converges.



My proof seems not yet completed. How do I continue it?










share|cite|improve this question
























  • You switch between $k$ and $n$ in your solution, you should fix that.
    – Jakobian
    Nov 22 at 14:50










  • I had edited it. Thanks for pointing my mistake.
    – weilam06
    Nov 22 at 15:02













up vote
4
down vote

favorite









up vote
4
down vote

favorite












Suppose the sequence ${a_n}_{n=1}^{infty}$ satisfies
$$midsumlimits^{n}_{k=1}{a_{k}}midleq Csqrt{n} spacespacespacespacespacespacespacespacespacespacespacespacespacespacespace n=1, 2, 3, cdots$$
where $C$ is a fixed (but arbitrary) number. Prove that the series
$$sumlimits^{infty}_{n=1}{frac{a_n}{n}}$$
converges.




My attempt: Suppose $b_n:= frac{1}{n}$; Abel's lemma on summation by parts gives
$$sumlimits^{k}_{n=1}{frac{a_n}{n}}=sumlimits^{k-1}_{n=1}{[sumlimits_{i=1}^{n}{a_i}cdot(b_n-b_{n+1})] + sumlimits_{i=1}^{k}{a_i}cdot b_k}$$



$$<sumlimits^{k-1}_{n=1}{midsumlimits_{i=1}^{n}{a_i}midcdot(b_n-b_{n+1}) +mid sumlimits_{i=1}^{k}{a_i}mid cdot b_k}$$



$$lesumlimits_{n=1}^{k}[Csqrt{n}cdot{(frac{1}{n}-frac{1}{n+1}})]+Csqrt{k}cdot frac{1}{k+1}$$



$$=sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}+frac{Csqrt{k}}{k+1}.$$



Since $krightarrowinfty$, therefore
$$frac{Csqrt{k}}{k+1}rightarrow 0.$$



Moreover, for the sigma notation, since
$$frac{Csqrt{n}}{n(n+1)}<frac{Csqrt{n}}{n^2}=frac{C}{n^frac{3}{2}}$$



Above is a $p$-series with $p=frac{3}{2}>1$, hence the series $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges. Even though $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges and $frac{Csqrt{k}}{k+1}$ approaches to $0$ for $krightarrowinfty$, but they do not imply the series $sumlimits^{infty}_{n=1}{frac{a_n}{n}}$ converges.



My proof seems not yet completed. How do I continue it?










share|cite|improve this question
















Suppose the sequence ${a_n}_{n=1}^{infty}$ satisfies
$$midsumlimits^{n}_{k=1}{a_{k}}midleq Csqrt{n} spacespacespacespacespacespacespacespacespacespacespacespacespacespacespace n=1, 2, 3, cdots$$
where $C$ is a fixed (but arbitrary) number. Prove that the series
$$sumlimits^{infty}_{n=1}{frac{a_n}{n}}$$
converges.




My attempt: Suppose $b_n:= frac{1}{n}$; Abel's lemma on summation by parts gives
$$sumlimits^{k}_{n=1}{frac{a_n}{n}}=sumlimits^{k-1}_{n=1}{[sumlimits_{i=1}^{n}{a_i}cdot(b_n-b_{n+1})] + sumlimits_{i=1}^{k}{a_i}cdot b_k}$$



$$<sumlimits^{k-1}_{n=1}{midsumlimits_{i=1}^{n}{a_i}midcdot(b_n-b_{n+1}) +mid sumlimits_{i=1}^{k}{a_i}mid cdot b_k}$$



$$lesumlimits_{n=1}^{k}[Csqrt{n}cdot{(frac{1}{n}-frac{1}{n+1}})]+Csqrt{k}cdot frac{1}{k+1}$$



$$=sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}+frac{Csqrt{k}}{k+1}.$$



Since $krightarrowinfty$, therefore
$$frac{Csqrt{k}}{k+1}rightarrow 0.$$



Moreover, for the sigma notation, since
$$frac{Csqrt{n}}{n(n+1)}<frac{Csqrt{n}}{n^2}=frac{C}{n^frac{3}{2}}$$



Above is a $p$-series with $p=frac{3}{2}>1$, hence the series $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges. Even though $sumlimits_{n=1}^{k}{frac{Csqrt{n}}{n(n+1)}}$ converges and $frac{Csqrt{k}}{k+1}$ approaches to $0$ for $krightarrowinfty$, but they do not imply the series $sumlimits^{infty}_{n=1}{frac{a_n}{n}}$ converges.



My proof seems not yet completed. How do I continue it?







sequences-and-series proof-verification convergence






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edited Nov 22 at 17:39









Martin Sleziak

44.4k7115268




44.4k7115268










asked Nov 22 at 14:41









weilam06

667




667












  • You switch between $k$ and $n$ in your solution, you should fix that.
    – Jakobian
    Nov 22 at 14:50










  • I had edited it. Thanks for pointing my mistake.
    – weilam06
    Nov 22 at 15:02


















  • You switch between $k$ and $n$ in your solution, you should fix that.
    – Jakobian
    Nov 22 at 14:50










  • I had edited it. Thanks for pointing my mistake.
    – weilam06
    Nov 22 at 15:02
















You switch between $k$ and $n$ in your solution, you should fix that.
– Jakobian
Nov 22 at 14:50




You switch between $k$ and $n$ in your solution, you should fix that.
– Jakobian
Nov 22 at 14:50












I had edited it. Thanks for pointing my mistake.
– weilam06
Nov 22 at 15:02




I had edited it. Thanks for pointing my mistake.
– weilam06
Nov 22 at 15:02










2 Answers
2






active

oldest

votes

















up vote
2
down vote













Let
$$
s_n=sum_{k=1}^na_n
$$

then $|s_n|le Csqrt{n}$ and
$$
begin{align}
sum_{k=1}^nfrac{a_k}k
&=sum_{k=1}^nfrac{s_k-s_{k-1}}k\
&=sum_{k=1}^nfrac{s_k}k-sum_{k=0}^{n-1}frac{s_k}{k+1}\
&=sum_{k=1}^ns_kleft(frac1k-frac1{k+1}right)+frac{s_n}{n+1}\
&=sum_{k=1}^nunderbrace{frac{s_k}{k(k+1)}}_{le Cfrac{sqrt{k}}{k^2}}+underbrace{ frac{s_n}{n+1} }_{le Cfrac{sqrt{n}}{nvphantom{k^2}}}
end{align}
$$

The sum converges and the extra term vanishes.





Comment brought into the answer



Since it does clarify the answer, I will bring the following comment into the answer.




The sequence of partial sums $sumlimits_{k=1}^nfrac{a_k}k$ is the same as the sequence $sumlimits_{k=1}^nfrac{s_k}{k(k+1)}+frac{s_n}{n+1}$ the latter sum converges absolutely, hence the partial sums are Cauchy and the extra term converges to $0$ hence it is Cauchy. The sum of two Cauchy sequences is Cauchy. Since the original sequence of partial sums is Cauchy, it converges.







share|cite|improve this answer























  • You only prove an upper bound for $|sum_{k=1}^n frac{a_k}{k}|$. The right-way to prove the convergence is to show that $sum_{k=1}^infty frac{a_k}{k}$ is a Cauchy-sequence!
    – p4sch
    Nov 22 at 14:56












  • Not exactly. Robjohn showed that the series converges absolutely.
    – Jakobian
    Nov 22 at 15:06










  • No, he didnt proved that the series converges absolutely! In general, it is false! Counterexample: $a_n = (-1)^n$! Problem: We dont have a bound for $sum_{k=1}^n |a_k|$ only for $sum_{k=1}^n a_k$.
    – p4sch
    Nov 22 at 15:07










  • @p4sch: the series in the last line converges absolutely by comparison to $sum k^{-3/2}$. That that series converges shows that the sequence of partial sums is Cauchy, and the series of partial sums are the same,
    – robjohn
    Nov 22 at 15:10










  • As I already said, you only prove that the sequence $(sum_{k=1}^n a_k/k)_{n in mathbb{N}}$ is bounded. Of course, this argument can be adapted in order to prove that this sequence is Cauchy! However, your answer doesn't prove convergence!
    – p4sch
    Nov 22 at 15:13


















up vote
1
down vote













In order to prove the convergence, it is enough to show that this sequence is a Cauchy-sequence (another way to prove the convergence is given by robjohn), because $mathbb{R}$, resp. $mathbb{C}$ is complete. Write $A_k := sum_{i=1}^k a_i$ and $B_k := sum_{i=1}^k a_i/i$. Note that $a_k = A_{k} - A_{k-1}$ Then for $n>m$
begin{align}
|B_n - B_m| = left| sum_{i=m+1}^n frac{a_i}{i} right| &= left| sum_{i=m+1}^n frac{A_i -A_{i-1}}{i} right| \
& = frac{|A_m|}{m+1} + frac{|A_n|}{n} + left| sum_{i=m+1}^{n-1} A_i left(frac{1}{i} - frac{1}{i-1} right) right|
end{align}

Since $(i^{-1}-(i-1)^{-1})= -(i(i-1))^{-1}$ Now we can use the bound to get the bound
$$frac{2C}{sqrt{m}}+ C sum_{i=m+1}^{n-1} frac{1}{(i-1) sqrt{i}}.$$
The last sum can be estimated by $int_{m-1}^infty frac{1}{x^{3/2}} d x < frac{3}{2} (m-1)^{-1/2}.$ All in all, we see that $$|B_n-B_m| le frac{3C}{(m-1)^{1/2}},$$
i.e. this is a Cauchy-sequence as claimed.



Note that this argument works also if we only require that $$|sum_{k=1}^n a_k| le C n^{1-varepsilon}.$$
Moreover, we cannot expect that $sum_{k=1}^infty a_k/k$ converges absolutely. Just take $a_k = (-1)^k$, then $$|sum_{k=1}^n a_k| le 1 le sqrt{n},$$ but $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$






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    2 Answers
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    up vote
    2
    down vote













    Let
    $$
    s_n=sum_{k=1}^na_n
    $$

    then $|s_n|le Csqrt{n}$ and
    $$
    begin{align}
    sum_{k=1}^nfrac{a_k}k
    &=sum_{k=1}^nfrac{s_k-s_{k-1}}k\
    &=sum_{k=1}^nfrac{s_k}k-sum_{k=0}^{n-1}frac{s_k}{k+1}\
    &=sum_{k=1}^ns_kleft(frac1k-frac1{k+1}right)+frac{s_n}{n+1}\
    &=sum_{k=1}^nunderbrace{frac{s_k}{k(k+1)}}_{le Cfrac{sqrt{k}}{k^2}}+underbrace{ frac{s_n}{n+1} }_{le Cfrac{sqrt{n}}{nvphantom{k^2}}}
    end{align}
    $$

    The sum converges and the extra term vanishes.





    Comment brought into the answer



    Since it does clarify the answer, I will bring the following comment into the answer.




    The sequence of partial sums $sumlimits_{k=1}^nfrac{a_k}k$ is the same as the sequence $sumlimits_{k=1}^nfrac{s_k}{k(k+1)}+frac{s_n}{n+1}$ the latter sum converges absolutely, hence the partial sums are Cauchy and the extra term converges to $0$ hence it is Cauchy. The sum of two Cauchy sequences is Cauchy. Since the original sequence of partial sums is Cauchy, it converges.







    share|cite|improve this answer























    • You only prove an upper bound for $|sum_{k=1}^n frac{a_k}{k}|$. The right-way to prove the convergence is to show that $sum_{k=1}^infty frac{a_k}{k}$ is a Cauchy-sequence!
      – p4sch
      Nov 22 at 14:56












    • Not exactly. Robjohn showed that the series converges absolutely.
      – Jakobian
      Nov 22 at 15:06










    • No, he didnt proved that the series converges absolutely! In general, it is false! Counterexample: $a_n = (-1)^n$! Problem: We dont have a bound for $sum_{k=1}^n |a_k|$ only for $sum_{k=1}^n a_k$.
      – p4sch
      Nov 22 at 15:07










    • @p4sch: the series in the last line converges absolutely by comparison to $sum k^{-3/2}$. That that series converges shows that the sequence of partial sums is Cauchy, and the series of partial sums are the same,
      – robjohn
      Nov 22 at 15:10










    • As I already said, you only prove that the sequence $(sum_{k=1}^n a_k/k)_{n in mathbb{N}}$ is bounded. Of course, this argument can be adapted in order to prove that this sequence is Cauchy! However, your answer doesn't prove convergence!
      – p4sch
      Nov 22 at 15:13















    up vote
    2
    down vote













    Let
    $$
    s_n=sum_{k=1}^na_n
    $$

    then $|s_n|le Csqrt{n}$ and
    $$
    begin{align}
    sum_{k=1}^nfrac{a_k}k
    &=sum_{k=1}^nfrac{s_k-s_{k-1}}k\
    &=sum_{k=1}^nfrac{s_k}k-sum_{k=0}^{n-1}frac{s_k}{k+1}\
    &=sum_{k=1}^ns_kleft(frac1k-frac1{k+1}right)+frac{s_n}{n+1}\
    &=sum_{k=1}^nunderbrace{frac{s_k}{k(k+1)}}_{le Cfrac{sqrt{k}}{k^2}}+underbrace{ frac{s_n}{n+1} }_{le Cfrac{sqrt{n}}{nvphantom{k^2}}}
    end{align}
    $$

    The sum converges and the extra term vanishes.





    Comment brought into the answer



    Since it does clarify the answer, I will bring the following comment into the answer.




    The sequence of partial sums $sumlimits_{k=1}^nfrac{a_k}k$ is the same as the sequence $sumlimits_{k=1}^nfrac{s_k}{k(k+1)}+frac{s_n}{n+1}$ the latter sum converges absolutely, hence the partial sums are Cauchy and the extra term converges to $0$ hence it is Cauchy. The sum of two Cauchy sequences is Cauchy. Since the original sequence of partial sums is Cauchy, it converges.







    share|cite|improve this answer























    • You only prove an upper bound for $|sum_{k=1}^n frac{a_k}{k}|$. The right-way to prove the convergence is to show that $sum_{k=1}^infty frac{a_k}{k}$ is a Cauchy-sequence!
      – p4sch
      Nov 22 at 14:56












    • Not exactly. Robjohn showed that the series converges absolutely.
      – Jakobian
      Nov 22 at 15:06










    • No, he didnt proved that the series converges absolutely! In general, it is false! Counterexample: $a_n = (-1)^n$! Problem: We dont have a bound for $sum_{k=1}^n |a_k|$ only for $sum_{k=1}^n a_k$.
      – p4sch
      Nov 22 at 15:07










    • @p4sch: the series in the last line converges absolutely by comparison to $sum k^{-3/2}$. That that series converges shows that the sequence of partial sums is Cauchy, and the series of partial sums are the same,
      – robjohn
      Nov 22 at 15:10










    • As I already said, you only prove that the sequence $(sum_{k=1}^n a_k/k)_{n in mathbb{N}}$ is bounded. Of course, this argument can be adapted in order to prove that this sequence is Cauchy! However, your answer doesn't prove convergence!
      – p4sch
      Nov 22 at 15:13













    up vote
    2
    down vote










    up vote
    2
    down vote









    Let
    $$
    s_n=sum_{k=1}^na_n
    $$

    then $|s_n|le Csqrt{n}$ and
    $$
    begin{align}
    sum_{k=1}^nfrac{a_k}k
    &=sum_{k=1}^nfrac{s_k-s_{k-1}}k\
    &=sum_{k=1}^nfrac{s_k}k-sum_{k=0}^{n-1}frac{s_k}{k+1}\
    &=sum_{k=1}^ns_kleft(frac1k-frac1{k+1}right)+frac{s_n}{n+1}\
    &=sum_{k=1}^nunderbrace{frac{s_k}{k(k+1)}}_{le Cfrac{sqrt{k}}{k^2}}+underbrace{ frac{s_n}{n+1} }_{le Cfrac{sqrt{n}}{nvphantom{k^2}}}
    end{align}
    $$

    The sum converges and the extra term vanishes.





    Comment brought into the answer



    Since it does clarify the answer, I will bring the following comment into the answer.




    The sequence of partial sums $sumlimits_{k=1}^nfrac{a_k}k$ is the same as the sequence $sumlimits_{k=1}^nfrac{s_k}{k(k+1)}+frac{s_n}{n+1}$ the latter sum converges absolutely, hence the partial sums are Cauchy and the extra term converges to $0$ hence it is Cauchy. The sum of two Cauchy sequences is Cauchy. Since the original sequence of partial sums is Cauchy, it converges.







    share|cite|improve this answer














    Let
    $$
    s_n=sum_{k=1}^na_n
    $$

    then $|s_n|le Csqrt{n}$ and
    $$
    begin{align}
    sum_{k=1}^nfrac{a_k}k
    &=sum_{k=1}^nfrac{s_k-s_{k-1}}k\
    &=sum_{k=1}^nfrac{s_k}k-sum_{k=0}^{n-1}frac{s_k}{k+1}\
    &=sum_{k=1}^ns_kleft(frac1k-frac1{k+1}right)+frac{s_n}{n+1}\
    &=sum_{k=1}^nunderbrace{frac{s_k}{k(k+1)}}_{le Cfrac{sqrt{k}}{k^2}}+underbrace{ frac{s_n}{n+1} }_{le Cfrac{sqrt{n}}{nvphantom{k^2}}}
    end{align}
    $$

    The sum converges and the extra term vanishes.





    Comment brought into the answer



    Since it does clarify the answer, I will bring the following comment into the answer.




    The sequence of partial sums $sumlimits_{k=1}^nfrac{a_k}k$ is the same as the sequence $sumlimits_{k=1}^nfrac{s_k}{k(k+1)}+frac{s_n}{n+1}$ the latter sum converges absolutely, hence the partial sums are Cauchy and the extra term converges to $0$ hence it is Cauchy. The sum of two Cauchy sequences is Cauchy. Since the original sequence of partial sums is Cauchy, it converges.








    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Nov 22 at 16:43

























    answered Nov 22 at 14:52









    robjohn

    263k27301622




    263k27301622












    • You only prove an upper bound for $|sum_{k=1}^n frac{a_k}{k}|$. The right-way to prove the convergence is to show that $sum_{k=1}^infty frac{a_k}{k}$ is a Cauchy-sequence!
      – p4sch
      Nov 22 at 14:56












    • Not exactly. Robjohn showed that the series converges absolutely.
      – Jakobian
      Nov 22 at 15:06










    • No, he didnt proved that the series converges absolutely! In general, it is false! Counterexample: $a_n = (-1)^n$! Problem: We dont have a bound for $sum_{k=1}^n |a_k|$ only for $sum_{k=1}^n a_k$.
      – p4sch
      Nov 22 at 15:07










    • @p4sch: the series in the last line converges absolutely by comparison to $sum k^{-3/2}$. That that series converges shows that the sequence of partial sums is Cauchy, and the series of partial sums are the same,
      – robjohn
      Nov 22 at 15:10










    • As I already said, you only prove that the sequence $(sum_{k=1}^n a_k/k)_{n in mathbb{N}}$ is bounded. Of course, this argument can be adapted in order to prove that this sequence is Cauchy! However, your answer doesn't prove convergence!
      – p4sch
      Nov 22 at 15:13


















    • You only prove an upper bound for $|sum_{k=1}^n frac{a_k}{k}|$. The right-way to prove the convergence is to show that $sum_{k=1}^infty frac{a_k}{k}$ is a Cauchy-sequence!
      – p4sch
      Nov 22 at 14:56












    • Not exactly. Robjohn showed that the series converges absolutely.
      – Jakobian
      Nov 22 at 15:06










    • No, he didnt proved that the series converges absolutely! In general, it is false! Counterexample: $a_n = (-1)^n$! Problem: We dont have a bound for $sum_{k=1}^n |a_k|$ only for $sum_{k=1}^n a_k$.
      – p4sch
      Nov 22 at 15:07










    • @p4sch: the series in the last line converges absolutely by comparison to $sum k^{-3/2}$. That that series converges shows that the sequence of partial sums is Cauchy, and the series of partial sums are the same,
      – robjohn
      Nov 22 at 15:10










    • As I already said, you only prove that the sequence $(sum_{k=1}^n a_k/k)_{n in mathbb{N}}$ is bounded. Of course, this argument can be adapted in order to prove that this sequence is Cauchy! However, your answer doesn't prove convergence!
      – p4sch
      Nov 22 at 15:13
















    You only prove an upper bound for $|sum_{k=1}^n frac{a_k}{k}|$. The right-way to prove the convergence is to show that $sum_{k=1}^infty frac{a_k}{k}$ is a Cauchy-sequence!
    – p4sch
    Nov 22 at 14:56






    You only prove an upper bound for $|sum_{k=1}^n frac{a_k}{k}|$. The right-way to prove the convergence is to show that $sum_{k=1}^infty frac{a_k}{k}$ is a Cauchy-sequence!
    – p4sch
    Nov 22 at 14:56














    Not exactly. Robjohn showed that the series converges absolutely.
    – Jakobian
    Nov 22 at 15:06




    Not exactly. Robjohn showed that the series converges absolutely.
    – Jakobian
    Nov 22 at 15:06












    No, he didnt proved that the series converges absolutely! In general, it is false! Counterexample: $a_n = (-1)^n$! Problem: We dont have a bound for $sum_{k=1}^n |a_k|$ only for $sum_{k=1}^n a_k$.
    – p4sch
    Nov 22 at 15:07




    No, he didnt proved that the series converges absolutely! In general, it is false! Counterexample: $a_n = (-1)^n$! Problem: We dont have a bound for $sum_{k=1}^n |a_k|$ only for $sum_{k=1}^n a_k$.
    – p4sch
    Nov 22 at 15:07












    @p4sch: the series in the last line converges absolutely by comparison to $sum k^{-3/2}$. That that series converges shows that the sequence of partial sums is Cauchy, and the series of partial sums are the same,
    – robjohn
    Nov 22 at 15:10




    @p4sch: the series in the last line converges absolutely by comparison to $sum k^{-3/2}$. That that series converges shows that the sequence of partial sums is Cauchy, and the series of partial sums are the same,
    – robjohn
    Nov 22 at 15:10












    As I already said, you only prove that the sequence $(sum_{k=1}^n a_k/k)_{n in mathbb{N}}$ is bounded. Of course, this argument can be adapted in order to prove that this sequence is Cauchy! However, your answer doesn't prove convergence!
    – p4sch
    Nov 22 at 15:13




    As I already said, you only prove that the sequence $(sum_{k=1}^n a_k/k)_{n in mathbb{N}}$ is bounded. Of course, this argument can be adapted in order to prove that this sequence is Cauchy! However, your answer doesn't prove convergence!
    – p4sch
    Nov 22 at 15:13










    up vote
    1
    down vote













    In order to prove the convergence, it is enough to show that this sequence is a Cauchy-sequence (another way to prove the convergence is given by robjohn), because $mathbb{R}$, resp. $mathbb{C}$ is complete. Write $A_k := sum_{i=1}^k a_i$ and $B_k := sum_{i=1}^k a_i/i$. Note that $a_k = A_{k} - A_{k-1}$ Then for $n>m$
    begin{align}
    |B_n - B_m| = left| sum_{i=m+1}^n frac{a_i}{i} right| &= left| sum_{i=m+1}^n frac{A_i -A_{i-1}}{i} right| \
    & = frac{|A_m|}{m+1} + frac{|A_n|}{n} + left| sum_{i=m+1}^{n-1} A_i left(frac{1}{i} - frac{1}{i-1} right) right|
    end{align}

    Since $(i^{-1}-(i-1)^{-1})= -(i(i-1))^{-1}$ Now we can use the bound to get the bound
    $$frac{2C}{sqrt{m}}+ C sum_{i=m+1}^{n-1} frac{1}{(i-1) sqrt{i}}.$$
    The last sum can be estimated by $int_{m-1}^infty frac{1}{x^{3/2}} d x < frac{3}{2} (m-1)^{-1/2}.$ All in all, we see that $$|B_n-B_m| le frac{3C}{(m-1)^{1/2}},$$
    i.e. this is a Cauchy-sequence as claimed.



    Note that this argument works also if we only require that $$|sum_{k=1}^n a_k| le C n^{1-varepsilon}.$$
    Moreover, we cannot expect that $sum_{k=1}^infty a_k/k$ converges absolutely. Just take $a_k = (-1)^k$, then $$|sum_{k=1}^n a_k| le 1 le sqrt{n},$$ but $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$






    share|cite|improve this answer



























      up vote
      1
      down vote













      In order to prove the convergence, it is enough to show that this sequence is a Cauchy-sequence (another way to prove the convergence is given by robjohn), because $mathbb{R}$, resp. $mathbb{C}$ is complete. Write $A_k := sum_{i=1}^k a_i$ and $B_k := sum_{i=1}^k a_i/i$. Note that $a_k = A_{k} - A_{k-1}$ Then for $n>m$
      begin{align}
      |B_n - B_m| = left| sum_{i=m+1}^n frac{a_i}{i} right| &= left| sum_{i=m+1}^n frac{A_i -A_{i-1}}{i} right| \
      & = frac{|A_m|}{m+1} + frac{|A_n|}{n} + left| sum_{i=m+1}^{n-1} A_i left(frac{1}{i} - frac{1}{i-1} right) right|
      end{align}

      Since $(i^{-1}-(i-1)^{-1})= -(i(i-1))^{-1}$ Now we can use the bound to get the bound
      $$frac{2C}{sqrt{m}}+ C sum_{i=m+1}^{n-1} frac{1}{(i-1) sqrt{i}}.$$
      The last sum can be estimated by $int_{m-1}^infty frac{1}{x^{3/2}} d x < frac{3}{2} (m-1)^{-1/2}.$ All in all, we see that $$|B_n-B_m| le frac{3C}{(m-1)^{1/2}},$$
      i.e. this is a Cauchy-sequence as claimed.



      Note that this argument works also if we only require that $$|sum_{k=1}^n a_k| le C n^{1-varepsilon}.$$
      Moreover, we cannot expect that $sum_{k=1}^infty a_k/k$ converges absolutely. Just take $a_k = (-1)^k$, then $$|sum_{k=1}^n a_k| le 1 le sqrt{n},$$ but $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$






      share|cite|improve this answer

























        up vote
        1
        down vote










        up vote
        1
        down vote









        In order to prove the convergence, it is enough to show that this sequence is a Cauchy-sequence (another way to prove the convergence is given by robjohn), because $mathbb{R}$, resp. $mathbb{C}$ is complete. Write $A_k := sum_{i=1}^k a_i$ and $B_k := sum_{i=1}^k a_i/i$. Note that $a_k = A_{k} - A_{k-1}$ Then for $n>m$
        begin{align}
        |B_n - B_m| = left| sum_{i=m+1}^n frac{a_i}{i} right| &= left| sum_{i=m+1}^n frac{A_i -A_{i-1}}{i} right| \
        & = frac{|A_m|}{m+1} + frac{|A_n|}{n} + left| sum_{i=m+1}^{n-1} A_i left(frac{1}{i} - frac{1}{i-1} right) right|
        end{align}

        Since $(i^{-1}-(i-1)^{-1})= -(i(i-1))^{-1}$ Now we can use the bound to get the bound
        $$frac{2C}{sqrt{m}}+ C sum_{i=m+1}^{n-1} frac{1}{(i-1) sqrt{i}}.$$
        The last sum can be estimated by $int_{m-1}^infty frac{1}{x^{3/2}} d x < frac{3}{2} (m-1)^{-1/2}.$ All in all, we see that $$|B_n-B_m| le frac{3C}{(m-1)^{1/2}},$$
        i.e. this is a Cauchy-sequence as claimed.



        Note that this argument works also if we only require that $$|sum_{k=1}^n a_k| le C n^{1-varepsilon}.$$
        Moreover, we cannot expect that $sum_{k=1}^infty a_k/k$ converges absolutely. Just take $a_k = (-1)^k$, then $$|sum_{k=1}^n a_k| le 1 le sqrt{n},$$ but $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$






        share|cite|improve this answer














        In order to prove the convergence, it is enough to show that this sequence is a Cauchy-sequence (another way to prove the convergence is given by robjohn), because $mathbb{R}$, resp. $mathbb{C}$ is complete. Write $A_k := sum_{i=1}^k a_i$ and $B_k := sum_{i=1}^k a_i/i$. Note that $a_k = A_{k} - A_{k-1}$ Then for $n>m$
        begin{align}
        |B_n - B_m| = left| sum_{i=m+1}^n frac{a_i}{i} right| &= left| sum_{i=m+1}^n frac{A_i -A_{i-1}}{i} right| \
        & = frac{|A_m|}{m+1} + frac{|A_n|}{n} + left| sum_{i=m+1}^{n-1} A_i left(frac{1}{i} - frac{1}{i-1} right) right|
        end{align}

        Since $(i^{-1}-(i-1)^{-1})= -(i(i-1))^{-1}$ Now we can use the bound to get the bound
        $$frac{2C}{sqrt{m}}+ C sum_{i=m+1}^{n-1} frac{1}{(i-1) sqrt{i}}.$$
        The last sum can be estimated by $int_{m-1}^infty frac{1}{x^{3/2}} d x < frac{3}{2} (m-1)^{-1/2}.$ All in all, we see that $$|B_n-B_m| le frac{3C}{(m-1)^{1/2}},$$
        i.e. this is a Cauchy-sequence as claimed.



        Note that this argument works also if we only require that $$|sum_{k=1}^n a_k| le C n^{1-varepsilon}.$$
        Moreover, we cannot expect that $sum_{k=1}^infty a_k/k$ converges absolutely. Just take $a_k = (-1)^k$, then $$|sum_{k=1}^n a_k| le 1 le sqrt{n},$$ but $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 22 at 15:25

























        answered Nov 22 at 15:15









        p4sch

        4,105216




        4,105216






























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