Mixing
If $sum_{n=1}^{infty}a_n$ converges then $sum_{n=1}^{infty}a_n+a_{n+1}+a_{n+2}$ converges
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I have the following problem:
Let $a_n$, $ninmathbb{N}$ be sequence
and let $b_n=a_n+a_{n+1}+a_{n+2}$
Prove that if $sum_{n=1}^{infty}a_n$ converges then $sum_{n=1}^{infty}b_n$ converges
My attempt:
Let $sum_{n=1}^{infty}a_n=a$
$sum_{n=1}^{infty}b_n=sum_{n=1}^{infty}a_n+a_{n+1}+a_{n+2}$=
$sum_{n=1}^{infty}a_n+sum_{n=1}^{infty}a_{n+1}+sum_{n=1}^{infty}a_{n+2}$=
$sum_{n=1}^{infty}a_n$+$sum_{n=1}^{infty}a_n-a_1$+$sum_{n=1}^{infty}a_n-a_1-a_2$=
$sum_{n=1}^{infty}3a_n-2a_1-a_2$
From the linearity of series we know that $sum_{n=1}^{infty}3a_n=3a$
And the series convergence isn't affected by a change in finite number of elements of the sum
So $sum_{n=1}^{infty}b_n$ converges
Is this any good? Is it sufficient?
real-analysis sequences-and-series convergence
asked Jan 22 at 20:23
ChangaChanga
234
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add a comment |
$begingroup$
I have the following problem:
Let $a_n$, $ninmathbb{N}$ be sequence
and let $b_n=a_n+a_{n+1}+a_{n+2}$
Prove that if $sum_{n=1}^{infty}a_n$ converges then $sum_{n=1}^{infty}b_n$ converges
My attempt:
Let $sum_{n=1}^{infty}a_n=a$
$sum_{n=1}^{infty}b_n=sum_{n=1}^{infty}a_n+a_{n+1}+a_{n+2}$=
$sum_{n=1}^{infty}a_n+sum_{n=1}^{infty}a_{n+1}+sum_{n=1}^{infty}a_{n+2}$=
$sum_{n=1}^{infty}a_n$+$sum_{n=1}^{infty}a_n-a_1$+$sum_{n=1}^{infty}a_n-a_1-a_2$=
$sum_{n=1}^{infty}3a_n-2a_1-a_2$
From the linearity of series we know that $sum_{n=1}^{infty}3a_n=3a$
And the series convergence isn't affected by a change in finite number of elements of the sum
So $sum_{n=1}^{infty}b_n$ converges
Is this any good? Is it sufficient?
real-analysis sequences-and-series convergence
asked Jan 22 at 20:23
ChangaChanga
234
$endgroup$
2
$begingroup$
Your proof is correct. Here's a more general proof: if $sum a_n, sum b_n$ converge, then $sum a_n + b_n$ converges and equals $sum a_n + sum b_n$. Use this and the fact that $sumlimits_{n=1}^infty a_{n+k}$ converges for every k if $sum a_n$ exists. You basically used these statements within your proof.
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– James Yang
Jan 22 at 20:32
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Yes your proof is correct
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– Mike
Jan 22 at 20:35
$begingroup$
Please put in parentheses.
$endgroup$
– zhw.
Jan 22 at 21:04
add a comment |
$begingroup$
I have the following problem:
Let $a_n$, $ninmathbb{N}$ be sequence
and let $b_n=a_n+a_{n+1}+a_{n+2}$
Prove that if $sum_{n=1}^{infty}a_n$ converges then $sum_{n=1}^{infty}b_n$ converges
My attempt:
Let $sum_{n=1}^{infty}a_n=a$
$sum_{n=1}^{infty}b_n=sum_{n=1}^{infty}a_n+a_{n+1}+a_{n+2}$=
$sum_{n=1}^{infty}a_n+sum_{n=1}^{infty}a_{n+1}+sum_{n=1}^{infty}a_{n+2}$=
$sum_{n=1}^{infty}a_n$+$sum_{n=1}^{infty}a_n-a_1$+$sum_{n=1}^{infty}a_n-a_1-a_2$=
$sum_{n=1}^{infty}3a_n-2a_1-a_2$
From the linearity of series we know that $sum_{n=1}^{infty}3a_n=3a$
And the series convergence isn't affected by a change in finite number of elements of the sum
So $sum_{n=1}^{infty}b_n$ converges
Is this any good? Is it sufficient?
real-analysis sequences-and-series convergence
asked Jan 22 at 20:23
ChangaChanga
234
$endgroup$
I have the following problem:
Let $a_n$, $ninmathbb{N}$ be sequence
and let $b_n=a_n+a_{n+1}+a_{n+2}$
Prove that if $sum_{n=1}^{infty}a_n$ converges then $sum_{n=1}^{infty}b_n$ converges
My attempt:
Let $sum_{n=1}^{infty}a_n=a$
$sum_{n=1}^{infty}b_n=sum_{n=1}^{infty}a_n+a_{n+1}+a_{n+2}$=
$sum_{n=1}^{infty}a_n+sum_{n=1}^{infty}a_{n+1}+sum_{n=1}^{infty}a_{n+2}$=
$sum_{n=1}^{infty}a_n$+$sum_{n=1}^{infty}a_n-a_1$+$sum_{n=1}^{infty}a_n-a_1-a_2$=
$sum_{n=1}^{infty}3a_n-2a_1-a_2$
From the linearity of series we know that $sum_{n=1}^{infty}3a_n=3a$
And the series convergence isn't affected by a change in finite number of elements of the sum
So $sum_{n=1}^{infty}b_n$ converges
Is this any good? Is it sufficient?
real-analysis sequences-and-series convergence
real-analysis sequences-and-series convergence
asked Jan 22 at 20:23
ChangaChanga
234
asked Jan 22 at 20:23
ChangaChanga
234
asked Jan 22 at 20:23
ChangaChanga
234
asked Jan 22 at 20:23
ChangaChanga
234
asked Jan 22 at 20:23
ChangaChanga
234
234
2
$begingroup$
Your proof is correct. Here's a more general proof: if $sum a_n, sum b_n$ converge, then $sum a_n + b_n$ converges and equals $sum a_n + sum b_n$. Use this and the fact that $sumlimits_{n=1}^infty a_{n+k}$ converges for every k if $sum a_n$ exists. You basically used these statements within your proof.
$endgroup$
– James Yang
Jan 22 at 20:32
$begingroup$
Yes your proof is correct
$endgroup$
– Mike
Jan 22 at 20:35
$begingroup$
Please put in parentheses.
$endgroup$
– zhw.
Jan 22 at 21:04
add a comment |
2
$begingroup$
Your proof is correct. Here's a more general proof: if $sum a_n, sum b_n$ converge, then $sum a_n + b_n$ converges and equals $sum a_n + sum b_n$. Use this and the fact that $sumlimits_{n=1}^infty a_{n+k}$ converges for every k if $sum a_n$ exists. You basically used these statements within your proof.
$endgroup$
– James Yang
Jan 22 at 20:32
$begingroup$
Yes your proof is correct
$endgroup$
– Mike
Jan 22 at 20:35
$begingroup$
Please put in parentheses.
$endgroup$
– zhw.
Jan 22 at 21:04
2
2
$begingroup$
Your proof is correct. Here's a more general proof: if $sum a_n, sum b_n$ converge, then $sum a_n + b_n$ converges and equals $sum a_n + sum b_n$. Use this and the fact that $sumlimits_{n=1}^infty a_{n+k}$ converges for every k if $sum a_n$ exists. You basically used these statements within your proof.
$endgroup$
– James Yang
Jan 22 at 20:32
$begingroup$
Your proof is correct. Here's a more general proof: if $sum a_n, sum b_n$ converge, then $sum a_n + b_n$ converges and equals $sum a_n + sum b_n$. Use this and the fact that $sumlimits_{n=1}^infty a_{n+k}$ converges for every k if $sum a_n$ exists. You basically used these statements within your proof.
$endgroup$
– James Yang
Jan 22 at 20:32
$begingroup$
Yes your proof is correct
$endgroup$
– Mike
Jan 22 at 20:35
$begingroup$
Yes your proof is correct
$endgroup$
– Mike
Jan 22 at 20:35
$begingroup$
Please put in parentheses.
$endgroup$
– zhw.
Jan 22 at 21:04
$begingroup$
Please put in parentheses.
$endgroup$
– zhw.
Jan 22 at 21:04
add a comment |
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$begingroup$
Your proof is correct. Here's a more general proof: if $sum a_n, sum b_n$ converge, then $sum a_n + b_n$ converges and equals $sum a_n + sum b_n$. Use this and the fact that $sumlimits_{n=1}^infty a_{n+k}$ converges for every k if $sum a_n$ exists. You basically used these statements within your proof.
$endgroup$
– James Yang
Jan 22 at 20:32
$begingroup$
Yes your proof is correct
$endgroup$
– Mike
Jan 22 at 20:35
$begingroup$
Please put in parentheses.
$endgroup$
– zhw.
Jan 22 at 21:04