Divergence of a sequence with floors and square roots Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Analyze the convergence or divergence of the following sequenceProving divergence of quotient sequenceSequence with convergent subsequences: divergent or convergent?sequence defined by $u_0=1/2$ and the recurrence relation $u_n+1=1-u_n^2$Confused about series and testing for convergence/divergence?complex subsequences of divergent sequencerelationship between convergence of a sequence and its corresponding series.If the sequence $lefta_nright$ has no two subsequence converging to two different limits then the sequence has a limit?Prove that $(-1)^n - n$ is divergent to $-infty$Find a divergent sequence whose distance lessens with each subsequent term
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Divergence of a sequence with floors and square roots
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Analyze the convergence or divergence of the following sequenceProving divergence of quotient sequenceSequence with convergent subsequences: divergent or convergent?sequence defined by $u_0=1/2$ and the recurrence relation $u_n+1=1-u_n^2$Confused about series and testing for convergence/divergence?complex subsequences of divergent sequencerelationship between convergence of a sequence and its corresponding series.If the sequence $lefta_nright$ has no two subsequence converging to two different limits then the sequence has a limit?Prove that $(-1)^n - n$ is divergent to $-infty$Find a divergent sequence whose distance lessens with each subsequent term
$begingroup$
Is there any specific approach to prove the divergence of a sequence? For example, I have this problem:
"Prove that the sequence $(a_n)_n ge 1$, where $a_n = nsqrt2 - [nsqrt2]
+ nsqrt3 -[nsqrt3]$ is divergent"
(the [ ] stands for the floor function )
I tried to solve it by finding to subsequences that have different limits, but it does not seem to work this way.
real-analysis sequences-and-series convergence divergence
asked Apr 1 at 13:55
DdangDdang
284
$endgroup$
add a comment |
$begingroup$
Is there any specific approach to prove the divergence of a sequence? For example, I have this problem:
"Prove that the sequence $(a_n)_n ge 1$, where $a_n = nsqrt2 - [nsqrt2]
+ nsqrt3 -[nsqrt3]$ is divergent"
(the [ ] stands for the floor function )
I tried to solve it by finding to subsequences that have different limits, but it does not seem to work this way.
real-analysis sequences-and-series convergence divergence
asked Apr 1 at 13:55
DdangDdang
284
$endgroup$
$begingroup$
Do you know that $ n sqrt2 pmod1 $ is dense in $[0,1]$?
$endgroup$
– Eric Towers
Apr 1 at 13:59
$begingroup$
I did not know that. But still I cannot solve the problem (using your fact), neither prove what you said.
$endgroup$
– Ddang
Apr 1 at 14:26
$begingroup$
You should try to prove that, for any constant $cin(0,1)$, there exist an infinite number of values of $n$ such that $nsqrt2-lfloornsqrt2rfloor>c$ or equivalently $nsqrt2 (bmod 1)$ is dense in $[0, 1]$. Since you have a sum consisting of an infinite number of terms greater than some constant, the sum diverges.
$endgroup$
– AlgorithmsX
Apr 1 at 14:36
$begingroup$
I tried but I still cannot prove it this way. If you find a solution, will you please post it?
$endgroup$
– Ddang
Apr 1 at 14:56
$begingroup$
math.stackexchange.com/a/189402/12042
$endgroup$
– Eric Towers
Apr 1 at 15:39
add a comment |
$begingroup$
Is there any specific approach to prove the divergence of a sequence? For example, I have this problem:
"Prove that the sequence $(a_n)_n ge 1$, where $a_n = nsqrt2 - [nsqrt2]
+ nsqrt3 -[nsqrt3]$ is divergent"
(the [ ] stands for the floor function )
I tried to solve it by finding to subsequences that have different limits, but it does not seem to work this way.
real-analysis sequences-and-series convergence divergence
asked Apr 1 at 13:55
DdangDdang
284
$endgroup$
Is there any specific approach to prove the divergence of a sequence? For example, I have this problem:
"Prove that the sequence $(a_n)_n ge 1$, where $a_n = nsqrt2 - [nsqrt2]
+ nsqrt3 -[nsqrt3]$ is divergent"
(the [ ] stands for the floor function )
I tried to solve it by finding to subsequences that have different limits, but it does not seem to work this way.
real-analysis sequences-and-series convergence divergence
real-analysis sequences-and-series convergence divergence
asked Apr 1 at 13:55
DdangDdang
284
asked Apr 1 at 13:55
DdangDdang
284
asked Apr 1 at 13:55
DdangDdang
284
asked Apr 1 at 13:55
DdangDdang
284
asked Apr 1 at 13:55
DdangDdang
284
284
$begingroup$
Do you know that $ n sqrt2 pmod1 $ is dense in $[0,1]$?
$endgroup$
– Eric Towers
Apr 1 at 13:59
$begingroup$
I did not know that. But still I cannot solve the problem (using your fact), neither prove what you said.
$endgroup$
– Ddang
Apr 1 at 14:26
$begingroup$
You should try to prove that, for any constant $cin(0,1)$, there exist an infinite number of values of $n$ such that $nsqrt2-lfloornsqrt2rfloor>c$ or equivalently $nsqrt2 (bmod 1)$ is dense in $[0, 1]$. Since you have a sum consisting of an infinite number of terms greater than some constant, the sum diverges.
$endgroup$
– AlgorithmsX
Apr 1 at 14:36
$begingroup$
I tried but I still cannot prove it this way. If you find a solution, will you please post it?
$endgroup$
– Ddang
Apr 1 at 14:56
$begingroup$
math.stackexchange.com/a/189402/12042
$endgroup$
– Eric Towers
Apr 1 at 15:39
add a comment |
$begingroup$
Do you know that $ n sqrt2 pmod1 $ is dense in $[0,1]$?
$endgroup$
– Eric Towers
Apr 1 at 13:59
$begingroup$
I did not know that. But still I cannot solve the problem (using your fact), neither prove what you said.
$endgroup$
– Ddang
Apr 1 at 14:26
$begingroup$
You should try to prove that, for any constant $cin(0,1)$, there exist an infinite number of values of $n$ such that $nsqrt2-lfloornsqrt2rfloor>c$ or equivalently $nsqrt2 (bmod 1)$ is dense in $[0, 1]$. Since you have a sum consisting of an infinite number of terms greater than some constant, the sum diverges.
$endgroup$
– AlgorithmsX
Apr 1 at 14:36
$begingroup$
I tried but I still cannot prove it this way. If you find a solution, will you please post it?
$endgroup$
– Ddang
Apr 1 at 14:56
$begingroup$
math.stackexchange.com/a/189402/12042
$endgroup$
– Eric Towers
Apr 1 at 15:39
$begingroup$
Do you know that $ n sqrt2 pmod1 $ is dense in $[0,1]$?
$endgroup$
– Eric Towers
Apr 1 at 13:59
$begingroup$
Do you know that $ n sqrt2 pmod1 $ is dense in $[0,1]$?
$endgroup$
– Eric Towers
Apr 1 at 13:59
$begingroup$
I did not know that. But still I cannot solve the problem (using your fact), neither prove what you said.
$endgroup$
– Ddang
Apr 1 at 14:26
$begingroup$
I did not know that. But still I cannot solve the problem (using your fact), neither prove what you said.
$endgroup$
– Ddang
Apr 1 at 14:26
$begingroup$
You should try to prove that, for any constant $cin(0,1)$, there exist an infinite number of values of $n$ such that $nsqrt2-lfloornsqrt2rfloor>c$ or equivalently $nsqrt2 (bmod 1)$ is dense in $[0, 1]$. Since you have a sum consisting of an infinite number of terms greater than some constant, the sum diverges.
$endgroup$
– AlgorithmsX
Apr 1 at 14:36
$begingroup$
You should try to prove that, for any constant $cin(0,1)$, there exist an infinite number of values of $n$ such that $nsqrt2-lfloornsqrt2rfloor>c$ or equivalently $nsqrt2 (bmod 1)$ is dense in $[0, 1]$. Since you have a sum consisting of an infinite number of terms greater than some constant, the sum diverges.
$endgroup$
– AlgorithmsX
Apr 1 at 14:36
$begingroup$
I tried but I still cannot prove it this way. If you find a solution, will you please post it?
$endgroup$
– Ddang
Apr 1 at 14:56
$begingroup$
I tried but I still cannot prove it this way. If you find a solution, will you please post it?
$endgroup$
– Ddang
Apr 1 at 14:56
$begingroup$
math.stackexchange.com/a/189402/12042
$endgroup$
– Eric Towers
Apr 1 at 15:39
$begingroup$
math.stackexchange.com/a/189402/12042
$endgroup$
– Eric Towers
Apr 1 at 15:39
add a comment |
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$begingroup$
Do you know that $ n sqrt2 pmod1 $ is dense in $[0,1]$?
$endgroup$
– Eric Towers
Apr 1 at 13:59
$begingroup$
I did not know that. But still I cannot solve the problem (using your fact), neither prove what you said.
$endgroup$
– Ddang
Apr 1 at 14:26
$begingroup$
You should try to prove that, for any constant $cin(0,1)$, there exist an infinite number of values of $n$ such that $nsqrt2-lfloornsqrt2rfloor>c$ or equivalently $nsqrt2 (bmod 1)$ is dense in $[0, 1]$. Since you have a sum consisting of an infinite number of terms greater than some constant, the sum diverges.
$endgroup$
– AlgorithmsX
Apr 1 at 14:36
$begingroup$
I tried but I still cannot prove it this way. If you find a solution, will you please post it?
$endgroup$
– Ddang
Apr 1 at 14:56
$begingroup$
math.stackexchange.com/a/189402/12042
$endgroup$
– Eric Towers
Apr 1 at 15:39