Mixing
Prove that if $sum_{n = 1}^{infty} a_{n}$ converges, then $sum_{k = 1}^{infty} A_{k}$ converges.
$begingroup$
Suppose that ${a_n}$ be a sequence of real numbers.let ${n_{k}}$ be an increasing sequence of positive integers, and let $A_{k} = a_{n_{k}} + a_{n_{k}+1}+ ... +a_{n_{k +1}-1}$, for each $k in mathbb{N}$. Prove that if $sum_{n = 1}^{infty} a_{n}$ converges, then $sum_{k = 1}^{infty} A_{k}$ converges. Give an example to show that the converse is not true.
Could anyone give me a hint please?especially I do not understand why $A_{k}$ looks like this and what does the author trying to teach me by this question? and why we are starting from $k = 1$ not $k =0$?
calculus sequences-and-series
asked Jan 13 at 15:54
hopefullyhopefully
240114
$endgroup$
add a comment |
$begingroup$
Suppose that ${a_n}$ be a sequence of real numbers.let ${n_{k}}$ be an increasing sequence of positive integers, and let $A_{k} = a_{n_{k}} + a_{n_{k}+1}+ ... +a_{n_{k +1}-1}$, for each $k in mathbb{N}$. Prove that if $sum_{n = 1}^{infty} a_{n}$ converges, then $sum_{k = 1}^{infty} A_{k}$ converges. Give an example to show that the converse is not true.
Could anyone give me a hint please?especially I do not understand why $A_{k}$ looks like this and what does the author trying to teach me by this question? and why we are starting from $k = 1$ not $k =0$?
calculus sequences-and-series
asked Jan 13 at 15:54
hopefullyhopefully
240114
$endgroup$
add a comment |
$begingroup$
Suppose that ${a_n}$ be a sequence of real numbers.let ${n_{k}}$ be an increasing sequence of positive integers, and let $A_{k} = a_{n_{k}} + a_{n_{k}+1}+ ... +a_{n_{k +1}-1}$, for each $k in mathbb{N}$. Prove that if $sum_{n = 1}^{infty} a_{n}$ converges, then $sum_{k = 1}^{infty} A_{k}$ converges. Give an example to show that the converse is not true.
Could anyone give me a hint please?especially I do not understand why $A_{k}$ looks like this and what does the author trying to teach me by this question? and why we are starting from $k = 1$ not $k =0$?
calculus sequences-and-series
asked Jan 13 at 15:54
hopefullyhopefully
240114
$endgroup$
Suppose that ${a_n}$ be a sequence of real numbers.let ${n_{k}}$ be an increasing sequence of positive integers, and let $A_{k} = a_{n_{k}} + a_{n_{k}+1}+ ... +a_{n_{k +1}-1}$, for each $k in mathbb{N}$. Prove that if $sum_{n = 1}^{infty} a_{n}$ converges, then $sum_{k = 1}^{infty} A_{k}$ converges. Give an example to show that the converse is not true.
Could anyone give me a hint please?especially I do not understand why $A_{k}$ looks like this and what does the author trying to teach me by this question? and why we are starting from $k = 1$ not $k =0$?
calculus sequences-and-series
calculus sequences-and-series
asked Jan 13 at 15:54
hopefullyhopefully
240114
asked Jan 13 at 15:54
hopefullyhopefully
240114
asked Jan 13 at 15:54
hopefullyhopefully
240114
asked Jan 13 at 15:54
hopefullyhopefully
240114
asked Jan 13 at 15:54
hopefullyhopefully
240114
240114
add a comment |
add a comment |
2 Answers
2
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oldest
votes
$begingroup$
HINT
Note that for $Kgeq1$, we have
$$sum_{k=1}^KA_k=sum_{n=n_1}^{n_{K+1}-1}a_n,$$
and use the Cauchy criterion for convergence. The exercise is trying to tell you that if the sum converges, then we can group consecutive terms together without affecting convergence.
As for the issue of $k=0$ vs. $k=1$, this is just a choice of style, depending on whether you say $0inmathbb N$ or $0notinmathbb N$.
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
$endgroup$
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:47
$begingroup$
Could you provide more details about using the Cauchy criterion please?
$endgroup$
– hopefully
Jan 15 at 6:54
add a comment |
$begingroup$
Hint: The sequence of partial sums of $sum_{k=1}^{infty}A_k$ is a subsequence of the sequence of partial sums of $sum_{n=n_1}^{infty}a_n.$
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
$endgroup$
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:46
$begingroup$
I think the second series starts from $n = 1$ not $n = n_{1}$ ? or this is not correct?
$endgroup$
– hopefully
Jan 15 at 8:24
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT
Note that for $Kgeq1$, we have
$$sum_{k=1}^KA_k=sum_{n=n_1}^{n_{K+1}-1}a_n,$$
and use the Cauchy criterion for convergence. The exercise is trying to tell you that if the sum converges, then we can group consecutive terms together without affecting convergence.
As for the issue of $k=0$ vs. $k=1$, this is just a choice of style, depending on whether you say $0inmathbb N$ or $0notinmathbb N$.
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
$endgroup$
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:47
$begingroup$
Could you provide more details about using the Cauchy criterion please?
$endgroup$
– hopefully
Jan 15 at 6:54
add a comment |
$begingroup$
HINT
Note that for $Kgeq1$, we have
$$sum_{k=1}^KA_k=sum_{n=n_1}^{n_{K+1}-1}a_n,$$
and use the Cauchy criterion for convergence. The exercise is trying to tell you that if the sum converges, then we can group consecutive terms together without affecting convergence.
As for the issue of $k=0$ vs. $k=1$, this is just a choice of style, depending on whether you say $0inmathbb N$ or $0notinmathbb N$.
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
$endgroup$
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:47
$begingroup$
Could you provide more details about using the Cauchy criterion please?
$endgroup$
– hopefully
Jan 15 at 6:54
add a comment |
$begingroup$
HINT
Note that for $Kgeq1$, we have
$$sum_{k=1}^KA_k=sum_{n=n_1}^{n_{K+1}-1}a_n,$$
and use the Cauchy criterion for convergence. The exercise is trying to tell you that if the sum converges, then we can group consecutive terms together without affecting convergence.
As for the issue of $k=0$ vs. $k=1$, this is just a choice of style, depending on whether you say $0inmathbb N$ or $0notinmathbb N$.
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
$endgroup$
HINT
Note that for $Kgeq1$, we have
$$sum_{k=1}^KA_k=sum_{n=n_1}^{n_{K+1}-1}a_n,$$
and use the Cauchy criterion for convergence. The exercise is trying to tell you that if the sum converges, then we can group consecutive terms together without affecting convergence.
As for the issue of $k=0$ vs. $k=1$, this is just a choice of style, depending on whether you say $0inmathbb N$ or $0notinmathbb N$.
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
edited Jan 13 at 17:29
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
answered Jan 13 at 16:13
AweyganAweygan
14.2k21441
14.2k21441
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:47
$begingroup$
Could you provide more details about using the Cauchy criterion please?
$endgroup$
– hopefully
Jan 15 at 6:54
add a comment |
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:47
$begingroup$
Could you provide more details about using the Cauchy criterion please?
$endgroup$
– hopefully
Jan 15 at 6:54
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:47
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:47
$begingroup$
Could you provide more details about using the Cauchy criterion please?
$endgroup$
– hopefully
Jan 15 at 6:54
$begingroup$
Could you provide more details about using the Cauchy criterion please?
$endgroup$
– hopefully
Jan 15 at 6:54
add a comment |
$begingroup$
Hint: The sequence of partial sums of $sum_{k=1}^{infty}A_k$ is a subsequence of the sequence of partial sums of $sum_{n=n_1}^{infty}a_n.$
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
$endgroup$
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:46
$begingroup$
I think the second series starts from $n = 1$ not $n = n_{1}$ ? or this is not correct?
$endgroup$
– hopefully
Jan 15 at 8:24
add a comment |
$begingroup$
Hint: The sequence of partial sums of $sum_{k=1}^{infty}A_k$ is a subsequence of the sequence of partial sums of $sum_{n=n_1}^{infty}a_n.$
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
$endgroup$
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:46
$begingroup$
I think the second series starts from $n = 1$ not $n = n_{1}$ ? or this is not correct?
$endgroup$
– hopefully
Jan 15 at 8:24
add a comment |
$begingroup$
Hint: The sequence of partial sums of $sum_{k=1}^{infty}A_k$ is a subsequence of the sequence of partial sums of $sum_{n=n_1}^{infty}a_n.$
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
$endgroup$
Hint: The sequence of partial sums of $sum_{k=1}^{infty}A_k$ is a subsequence of the sequence of partial sums of $sum_{n=n_1}^{infty}a_n.$
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
answered Jan 13 at 18:02
zhw.zhw.
72.9k43175
72.9k43175
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:46
$begingroup$
I think the second series starts from $n = 1$ not $n = n_{1}$ ? or this is not correct?
$endgroup$
– hopefully
Jan 15 at 8:24
add a comment |
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:46
$begingroup$
I think the second series starts from $n = 1$ not $n = n_{1}$ ? or this is not correct?
$endgroup$
– hopefully
Jan 15 at 8:24
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:46
$begingroup$
Could you please give a numeric example for me for the two serieses please?
$endgroup$
– hopefully
Jan 15 at 6:46
$begingroup$
I think the second series starts from $n = 1$ not $n = n_{1}$ ? or this is not correct?
$endgroup$
– hopefully
Jan 15 at 8:24
$begingroup$
I think the second series starts from $n = 1$ not $n = n_{1}$ ? or this is not correct?
$endgroup$
– hopefully
Jan 15 at 8:24
add a comment |
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