If $ sum_{n=1}^{infty} a_n $ is absolutely convergent, then every rearrangement of $ sum_{n=1}^{infty} a_n $...
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This question was asked before but my proof is different from that proof.
So will you please check my proof?
Let $ sum_{n=1}^{infty} a_n $ be an absolutely convergent series then every partial sum of $ sum_{n=1}^{infty} |a_n| $ is bounded by some $M>0$. Let $ sum_{k=1}^{infty} b_k $ be a rearrangement of $ sum_{n=1}^{infty} a_n $.
Now consider the kth partial sum of $ sum_{k=1}^{infty} |b_k| $ that is $ s_k = |b_1|+|b_2|+cdots+|b_k| $ since both the sequences $ (a_n) $ and $ ( b_n) $ have same elements but in different arrangement so we can write $ s_k=|a_{p_1}|+|a_{p_2}|+cdots+|a_{p_k}| $. Let the greatest index be $p_j$, where $ 1le j le k $, then $ s_k le |a_1|+|a_2|+cdots+|a_{p_j}| lt M$. Hence the proof.
real-analysis proof-verification
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
asked Nov 27 at 8:10
suchanda adhikari
516
add a comment |
up vote
0
down vote
favorite
This question was asked before but my proof is different from that proof.
So will you please check my proof?
Let $ sum_{n=1}^{infty} a_n $ be an absolutely convergent series then every partial sum of $ sum_{n=1}^{infty} |a_n| $ is bounded by some $M>0$. Let $ sum_{k=1}^{infty} b_k $ be a rearrangement of $ sum_{n=1}^{infty} a_n $.
Now consider the kth partial sum of $ sum_{k=1}^{infty} |b_k| $ that is $ s_k = |b_1|+|b_2|+cdots+|b_k| $ since both the sequences $ (a_n) $ and $ ( b_n) $ have same elements but in different arrangement so we can write $ s_k=|a_{p_1}|+|a_{p_2}|+cdots+|a_{p_k}| $. Let the greatest index be $p_j$, where $ 1le j le k $, then $ s_k le |a_1|+|a_2|+cdots+|a_{p_j}| lt M$. Hence the proof.
real-analysis proof-verification
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
asked Nov 27 at 8:10
suchanda adhikari
516
1
Yes, this is correct.
– Kavi Rama Murthy
Nov 27 at 8:12
1
Maybe simpler to say that if the sum is bounded, the sum of any subset is also bounded.
– Yves Daoust
Nov 27 at 8:23
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This question was asked before but my proof is different from that proof.
So will you please check my proof?
Let $ sum_{n=1}^{infty} a_n $ be an absolutely convergent series then every partial sum of $ sum_{n=1}^{infty} |a_n| $ is bounded by some $M>0$. Let $ sum_{k=1}^{infty} b_k $ be a rearrangement of $ sum_{n=1}^{infty} a_n $.
Now consider the kth partial sum of $ sum_{k=1}^{infty} |b_k| $ that is $ s_k = |b_1|+|b_2|+cdots+|b_k| $ since both the sequences $ (a_n) $ and $ ( b_n) $ have same elements but in different arrangement so we can write $ s_k=|a_{p_1}|+|a_{p_2}|+cdots+|a_{p_k}| $. Let the greatest index be $p_j$, where $ 1le j le k $, then $ s_k le |a_1|+|a_2|+cdots+|a_{p_j}| lt M$. Hence the proof.
real-analysis proof-verification
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
asked Nov 27 at 8:10
suchanda adhikari
516
This question was asked before but my proof is different from that proof.
So will you please check my proof?
Let $ sum_{n=1}^{infty} a_n $ be an absolutely convergent series then every partial sum of $ sum_{n=1}^{infty} |a_n| $ is bounded by some $M>0$. Let $ sum_{k=1}^{infty} b_k $ be a rearrangement of $ sum_{n=1}^{infty} a_n $.
Now consider the kth partial sum of $ sum_{k=1}^{infty} |b_k| $ that is $ s_k = |b_1|+|b_2|+cdots+|b_k| $ since both the sequences $ (a_n) $ and $ ( b_n) $ have same elements but in different arrangement so we can write $ s_k=|a_{p_1}|+|a_{p_2}|+cdots+|a_{p_k}| $. Let the greatest index be $p_j$, where $ 1le j le k $, then $ s_k le |a_1|+|a_2|+cdots+|a_{p_j}| lt M$. Hence the proof.
real-analysis proof-verification
real-analysis proof-verification
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
asked Nov 27 at 8:10
suchanda adhikari
516
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
asked Nov 27 at 8:10
suchanda adhikari
516
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
edited Nov 27 at 8:28
José Carlos Santos
147k22117218
147k22117218
asked Nov 27 at 8:10
suchanda adhikari
516
asked Nov 27 at 8:10
suchanda adhikari
516
asked Nov 27 at 8:10
suchanda adhikari
516
516
1
Yes, this is correct.
– Kavi Rama Murthy
Nov 27 at 8:12
1
Maybe simpler to say that if the sum is bounded, the sum of any subset is also bounded.
– Yves Daoust
Nov 27 at 8:23
add a comment |
1
Yes, this is correct.
– Kavi Rama Murthy
Nov 27 at 8:12
1
Maybe simpler to say that if the sum is bounded, the sum of any subset is also bounded.
– Yves Daoust
Nov 27 at 8:23
1
1
Yes, this is correct.
– Kavi Rama Murthy
Nov 27 at 8:12
Yes, this is correct.
– Kavi Rama Murthy
Nov 27 at 8:12
1
1
Maybe simpler to say that if the sum is bounded, the sum of any subset is also bounded.
– Yves Daoust
Nov 27 at 8:23
Maybe simpler to say that if the sum is bounded, the sum of any subset is also bounded.
– Yves Daoust
Nov 27 at 8:23
add a comment |
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1
Yes, this is correct.
– Kavi Rama Murthy
Nov 27 at 8:12
1
Maybe simpler to say that if the sum is bounded, the sum of any subset is also bounded.
– Yves Daoust
Nov 27 at 8:23