Mixing
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Upper bound on some sum
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Let $ngeq 1$, $epsilon>0$, $vin mathbbR^n$ and $c>0$. I want to prove that:
$$sum_k=1^n left|v^2_k+1-v^2_k right|sqrtepsilon frackn^frac13+c leq sqrtepsilonn^frac13sum_k=1^n (v_k+1-v_k)^2 + frac1sqrtepsilon n^frac13sum_k=1^n left(epsilon frackn^frac13+cright)v^2_k .$$
This is from a textbook I am currently studying. I understand that we get the v's on the RHS thanks to $v_k+1^2-v_k^2 leq (v_k+1-v_k)^2 + 2v_k+1v_k $ and Cauchy-Schwarz (applied on $v_k+1v_k$) but I am struggling for the whole bound.
real-analysis summation upper-lower-bounds
asked Jul 19 at 17:10
anonymus
854312
add a comment |Â
up vote
0
down vote
favorite
Let $ngeq 1$, $epsilon>0$, $vin mathbbR^n$ and $c>0$. I want to prove that:
$$sum_k=1^n left|v^2_k+1-v^2_k right|sqrtepsilon frackn^frac13+c leq sqrtepsilonn^frac13sum_k=1^n (v_k+1-v_k)^2 + frac1sqrtepsilon n^frac13sum_k=1^n left(epsilon frackn^frac13+cright)v^2_k .$$
This is from a textbook I am currently studying. I understand that we get the v's on the RHS thanks to $v_k+1^2-v_k^2 leq (v_k+1-v_k)^2 + 2v_k+1v_k $ and Cauchy-Schwarz (applied on $v_k+1v_k$) but I am struggling for the whole bound.
real-analysis summation upper-lower-bounds
asked Jul 19 at 17:10
anonymus
854312
How can $k$ range from $1$ to $n$ for $|v_k+1-v_k|^2$ and $(v_k+1-v_k)^2$ when you only know values for $v_1, ldots, v_n$?
– Tengu
Jul 20 at 1:07
@Tengu by convention, one sets $v_0=v_n+1=0$.
– anonymus
Jul 20 at 17:48
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $ngeq 1$, $epsilon>0$, $vin mathbbR^n$ and $c>0$. I want to prove that:
$$sum_k=1^n left|v^2_k+1-v^2_k right|sqrtepsilon frackn^frac13+c leq sqrtepsilonn^frac13sum_k=1^n (v_k+1-v_k)^2 + frac1sqrtepsilon n^frac13sum_k=1^n left(epsilon frackn^frac13+cright)v^2_k .$$
This is from a textbook I am currently studying. I understand that we get the v's on the RHS thanks to $v_k+1^2-v_k^2 leq (v_k+1-v_k)^2 + 2v_k+1v_k $ and Cauchy-Schwarz (applied on $v_k+1v_k$) but I am struggling for the whole bound.
real-analysis summation upper-lower-bounds
asked Jul 19 at 17:10
anonymus
854312
Let $ngeq 1$, $epsilon>0$, $vin mathbbR^n$ and $c>0$. I want to prove that:
$$sum_k=1^n left|v^2_k+1-v^2_k right|sqrtepsilon frackn^frac13+c leq sqrtepsilonn^frac13sum_k=1^n (v_k+1-v_k)^2 + frac1sqrtepsilon n^frac13sum_k=1^n left(epsilon frackn^frac13+cright)v^2_k .$$
This is from a textbook I am currently studying. I understand that we get the v's on the RHS thanks to $v_k+1^2-v_k^2 leq (v_k+1-v_k)^2 + 2v_k+1v_k $ and Cauchy-Schwarz (applied on $v_k+1v_k$) but I am struggling for the whole bound.
real-analysis summation upper-lower-bounds
asked Jul 19 at 17:10
anonymus
854312
asked Jul 19 at 17:10
anonymus
854312
asked Jul 19 at 17:10
anonymus
854312
asked Jul 19 at 17:10
anonymus
854312
854312
How can $k$ range from $1$ to $n$ for $|v_k+1-v_k|^2$ and $(v_k+1-v_k)^2$ when you only know values for $v_1, ldots, v_n$?
– Tengu
Jul 20 at 1:07
@Tengu by convention, one sets $v_0=v_n+1=0$.
– anonymus
Jul 20 at 17:48
add a comment |Â
How can $k$ range from $1$ to $n$ for $|v_k+1-v_k|^2$ and $(v_k+1-v_k)^2$ when you only know values for $v_1, ldots, v_n$?
– Tengu
Jul 20 at 1:07
@Tengu by convention, one sets $v_0=v_n+1=0$.
– anonymus
Jul 20 at 17:48
How can $k$ range from $1$ to $n$ for $|v_k+1-v_k|^2$ and $(v_k+1-v_k)^2$ when you only know values for $v_1, ldots, v_n$?
– Tengu
Jul 20 at 1:07
How can $k$ range from $1$ to $n$ for $|v_k+1-v_k|^2$ and $(v_k+1-v_k)^2$ when you only know values for $v_1, ldots, v_n$?
– Tengu
Jul 20 at 1:07
@Tengu by convention, one sets $v_0=v_n+1=0$.
– anonymus
Jul 20 at 17:48
@Tengu by convention, one sets $v_0=v_n+1=0$.
– anonymus
Jul 20 at 17:48
add a comment |Â
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How can $k$ range from $1$ to $n$ for $|v_k+1-v_k|^2$ and $(v_k+1-v_k)^2$ when you only know values for $v_1, ldots, v_n$?
– Tengu
Jul 20 at 1:07
@Tengu by convention, one sets $v_0=v_n+1=0$.
– anonymus
Jul 20 at 17:48