If $|f(p+q)-f(q)|leq p/q$ for rational $p$ and $q$ (with $qneq 0$), then $sum_{i=0}^k |f(2^k) -f(2^i)| leq...
$begingroup$
If $left|f(p+q) - f(q)right|leq p/q$ for all rational $p$ and $q$, with $q neq 0$, then prove that
$$sum_{i=0}^k left|fleft(2^kright) -fleft(2^iright)right| leq frac12 k(k-1)$$
My try:
I consider the sum for $i=r $ which gives the inequality from given property of function $$ left |fleft(2^kright) -fleft(2^iright)right| leq 2^{k-i} - 1$$, and then summed it, but it doesn't work.
calculus functions summation
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
$endgroup$
add a comment |
$begingroup$
If $left|f(p+q) - f(q)right|leq p/q$ for all rational $p$ and $q$, with $q neq 0$, then prove that
$$sum_{i=0}^k left|fleft(2^kright) -fleft(2^iright)right| leq frac12 k(k-1)$$
My try:
I consider the sum for $i=r $ which gives the inequality from given property of function $$ left |fleft(2^kright) -fleft(2^iright)right| leq 2^{k-i} - 1$$, and then summed it, but it doesn't work.
calculus functions summation
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
$endgroup$
$begingroup$
Thanks for this
$endgroup$
– Keshav Sharma
Jan 3 at 5:24
add a comment |
$begingroup$
If $left|f(p+q) - f(q)right|leq p/q$ for all rational $p$ and $q$, with $q neq 0$, then prove that
$$sum_{i=0}^k left|fleft(2^kright) -fleft(2^iright)right| leq frac12 k(k-1)$$
My try:
I consider the sum for $i=r $ which gives the inequality from given property of function $$ left |fleft(2^kright) -fleft(2^iright)right| leq 2^{k-i} - 1$$, and then summed it, but it doesn't work.
calculus functions summation
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
$endgroup$
If $left|f(p+q) - f(q)right|leq p/q$ for all rational $p$ and $q$, with $q neq 0$, then prove that
$$sum_{i=0}^k left|fleft(2^kright) -fleft(2^iright)right| leq frac12 k(k-1)$$
My try:
I consider the sum for $i=r $ which gives the inequality from given property of function $$ left |fleft(2^kright) -fleft(2^iright)right| leq 2^{k-i} - 1$$, and then summed it, but it doesn't work.
calculus functions summation
calculus functions summation
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
edited Jan 3 at 10:18
Keshav Sharma
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
asked Jan 2 at 16:15
Keshav SharmaKeshav Sharma
986
986
$begingroup$
Thanks for this
$endgroup$
– Keshav Sharma
Jan 3 at 5:24
add a comment |
$begingroup$
Thanks for this
$endgroup$
– Keshav Sharma
Jan 3 at 5:24
$begingroup$
Thanks for this
$endgroup$
– Keshav Sharma
Jan 3 at 5:24
$begingroup$
Thanks for this
$endgroup$
– Keshav Sharma
Jan 3 at 5:24
add a comment |
1 Answer
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$begingroup$
By triangle inequality
$|f(2^k)-f(2^j)| leq |f(2^k)-f(2^{k-1})|+...|f(2^{j+1})-f(2^j)|leq k-j$
Hence $sum_{i=0}^{i=k}{|f(2^k)-f(2^i)|} leq sum_{i=0}^{i=k}{k-i} leq 0.5(k)(k+1)$
Still not as good as $0.5(k)(k-1)$
answered Jan 3 at 11:18
acreativenameacreativename
7617
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
By triangle inequality
$|f(2^k)-f(2^j)| leq |f(2^k)-f(2^{k-1})|+...|f(2^{j+1})-f(2^j)|leq k-j$
Hence $sum_{i=0}^{i=k}{|f(2^k)-f(2^i)|} leq sum_{i=0}^{i=k}{k-i} leq 0.5(k)(k+1)$
Still not as good as $0.5(k)(k-1)$
answered Jan 3 at 11:18
acreativenameacreativename
7617
$endgroup$
add a comment |
$begingroup$
By triangle inequality
$|f(2^k)-f(2^j)| leq |f(2^k)-f(2^{k-1})|+...|f(2^{j+1})-f(2^j)|leq k-j$
Hence $sum_{i=0}^{i=k}{|f(2^k)-f(2^i)|} leq sum_{i=0}^{i=k}{k-i} leq 0.5(k)(k+1)$
Still not as good as $0.5(k)(k-1)$
answered Jan 3 at 11:18
acreativenameacreativename
7617
$endgroup$
add a comment |
$begingroup$
By triangle inequality
$|f(2^k)-f(2^j)| leq |f(2^k)-f(2^{k-1})|+...|f(2^{j+1})-f(2^j)|leq k-j$
Hence $sum_{i=0}^{i=k}{|f(2^k)-f(2^i)|} leq sum_{i=0}^{i=k}{k-i} leq 0.5(k)(k+1)$
Still not as good as $0.5(k)(k-1)$
answered Jan 3 at 11:18
acreativenameacreativename
7617
$endgroup$
By triangle inequality
$|f(2^k)-f(2^j)| leq |f(2^k)-f(2^{k-1})|+...|f(2^{j+1})-f(2^j)|leq k-j$
Hence $sum_{i=0}^{i=k}{|f(2^k)-f(2^i)|} leq sum_{i=0}^{i=k}{k-i} leq 0.5(k)(k+1)$
Still not as good as $0.5(k)(k-1)$
answered Jan 3 at 11:18
acreativenameacreativename
7617
answered Jan 3 at 11:18
acreativenameacreativename
7617
answered Jan 3 at 11:18
acreativenameacreativename
7617
answered Jan 3 at 11:18
acreativenameacreativename
7617
7617
add a comment |
add a comment |
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$begingroup$
Thanks for this
$endgroup$
– Keshav Sharma
Jan 3 at 5:24