Show the series $sumlimits^{infty}_{n=1}{frac{a_n}{n}}$ converges
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?
sequences-and-series proof-verification convergence
add a comment |
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?
sequences-and-series proof-verification convergence
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
add a comment |
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?
sequences-and-series proof-verification convergence
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
sequences-and-series proof-verification convergence
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
add a comment |
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
add a comment |
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.
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
|
show 2 more comments
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 $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
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.
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
|
show 2 more comments
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.
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
|
show 2 more comments
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.
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.
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
|
show 2 more comments
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
|
show 2 more comments
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 $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$
add a comment |
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 $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$
add a comment |
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 $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$
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 $$sum_{k=1}^infty frac{|a_k|}{k} = sum_{k=1}^infty frac{1}{k} =infty.$$
edited Nov 22 at 15:25
answered Nov 22 at 15:15
p4sch
4,105216
4,105216
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009219%2fshow-the-series-sum-limits-infty-n-1-fraca-nn-converges%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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