Steps to determine the interval of continuity of $f(x)= sum_n=2^infty frac(sinnx)^2sqrtn$Does $sum_n=1^infty frac(-1)^fracn(n-1)2sqrt[3]n^2+e^x $ uniformly converge in $(-infty, infty)$?Does $sum_n=1^infty frac sin(fracnpi6)sqrt n^4 + 1$ converge?How to prove the divergence/convergence of the following series: $sum_n=0^infty fracsqrt[3]nsqrt[3]n^4-1$?Determine whether $sum_n=1^infty frac sin(n)n$ convergesInterval of convergence of $sum_n=1^infty x^ln(n)$.Does the series converge: $sum_2^inftyfraclog nn(log log n)^2$Does the series $sum_k=1^inftyleft(sqrtk+frac1k-sqrtkright)$ converge or diverge?Determine if $sum_n=1^infty fracn^2n^3+3 $ converges or divergesDetermine the convergence/divergence of $sum_n=1^inftyfraclnn!n^3$Does the series $sum_n=1^infty fracnsqrt[3]8n^5-1$ Converge?
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Steps to determine the interval of continuity of $f(x)= sum_n=2^infty frac(sinnx)^2sqrtn$
Does $sum_n=1^infty frac(-1)^fracn(n-1)2sqrt[3]n^2+e^x $ uniformly converge in $(-infty, infty)$?Does $sum_n=1^infty frac sin(fracnpi6)sqrt n^4 + 1$ converge?How to prove the divergence/convergence of the following series: $sum_n=0^infty fracsqrt[3]nsqrt[3]n^4-1$?Determine whether $sum_n=1^infty frac (-1)^nn$ convergesInterval of convergence of $sum_n=1^infty x^ln(n)$.Does the series converge: $sum_2^inftyfraclog nn(log log n)^2$Does the series $sum_k=1^inftyleft(sqrtk+frac1k-sqrtkright)$ converge or diverge?Determine if $sum_n=1^infty fracn^2n^3+3 $ converges or divergesDetermine the convergence/divergence of $sum_n=1^inftyfraclnn!n^3$Does the series $sum_n=1^infty fracnsqrt[3]8n^5-1$ Converge?
$begingroup$
I am trying I determine the interval of continuity of $$f_n(x)= sum_n=2^infty frac(sinnx)^2sqrtn$$
I tried to find the domain by Dirichlet's test, but the sum $sum _n=1^infty (sin (nx)^2)$ does not converge. I also tried the ratio test on which I got stuck on. The root test was inconclusive. What other tests can I apply?
real-analysis calculus sequences-and-series convergence continuity
asked Mar 12 at 20:45
E.NoleE.Nole
192114
$endgroup$
add a comment |
$begingroup$
I am trying I determine the interval of continuity of $$f_n(x)= sum_n=2^infty frac(sinnx)^2sqrtn$$
I tried to find the domain by Dirichlet's test, but the sum $sum _n=1^infty (sin (nx)^2)$ does not converge. I also tried the ratio test on which I got stuck on. The root test was inconclusive. What other tests can I apply?
real-analysis calculus sequences-and-series convergence continuity
asked Mar 12 at 20:45
E.NoleE.Nole
192114
$endgroup$
$begingroup$
First, the function does not depends on $n$. It should be $f$. Secondly, have you determined the definition domain of your function ?
$endgroup$
– TheSilverDoe
Mar 12 at 20:53
$begingroup$
@TheSilverDoe I was trying to find the domain using Dirichlet's test, the root test and the ratio test but didn't get anything helpful. I initially guessed that the domain is all real $x$ except $0$, but I can't back my argument yet.
$endgroup$
– E.Nole
Mar 12 at 20:58
$begingroup$
Unfortunately, I think that $f$ is not defined on any interval...
$endgroup$
– TheSilverDoe
Mar 12 at 21:02
$begingroup$
@TheSilverDoe you mean the series diverges for all $x$?
$endgroup$
– E.Nole
Mar 12 at 21:06
add a comment |
$begingroup$
I am trying I determine the interval of continuity of $$f_n(x)= sum_n=2^infty frac(sinnx)^2sqrtn$$
I tried to find the domain by Dirichlet's test, but the sum $sum _n=1^infty (sin (nx)^2)$ does not converge. I also tried the ratio test on which I got stuck on. The root test was inconclusive. What other tests can I apply?
real-analysis calculus sequences-and-series convergence continuity
asked Mar 12 at 20:45
E.NoleE.Nole
192114
$endgroup$
I am trying I determine the interval of continuity of $$f_n(x)= sum_n=2^infty frac(sinnx)^2sqrtn$$
I tried to find the domain by Dirichlet's test, but the sum $sum _n=1^infty (sin (nx)^2)$ does not converge. I also tried the ratio test on which I got stuck on. The root test was inconclusive. What other tests can I apply?
real-analysis calculus sequences-and-series convergence continuity
real-analysis calculus sequences-and-series convergence continuity
asked Mar 12 at 20:45
E.NoleE.Nole
192114
asked Mar 12 at 20:45
E.NoleE.Nole
192114
edited Mar 12 at 20:55
E.Nole
asked Mar 12 at 20:45
E.NoleE.Nole
192114
asked Mar 12 at 20:45
E.NoleE.Nole
192114
asked Mar 12 at 20:45
E.NoleE.Nole
192114
192114
$begingroup$
First, the function does not depends on $n$. It should be $f$. Secondly, have you determined the definition domain of your function ?
$endgroup$
– TheSilverDoe
Mar 12 at 20:53
$begingroup$
@TheSilverDoe I was trying to find the domain using Dirichlet's test, the root test and the ratio test but didn't get anything helpful. I initially guessed that the domain is all real $x$ except $0$, but I can't back my argument yet.
$endgroup$
– E.Nole
Mar 12 at 20:58
$begingroup$
Unfortunately, I think that $f$ is not defined on any interval...
$endgroup$
– TheSilverDoe
Mar 12 at 21:02
$begingroup$
@TheSilverDoe you mean the series diverges for all $x$?
$endgroup$
– E.Nole
Mar 12 at 21:06
add a comment |
$begingroup$
First, the function does not depends on $n$. It should be $f$. Secondly, have you determined the definition domain of your function ?
$endgroup$
– TheSilverDoe
Mar 12 at 20:53
$begingroup$
@TheSilverDoe I was trying to find the domain using Dirichlet's test, the root test and the ratio test but didn't get anything helpful. I initially guessed that the domain is all real $x$ except $0$, but I can't back my argument yet.
$endgroup$
– E.Nole
Mar 12 at 20:58
$begingroup$
Unfortunately, I think that $f$ is not defined on any interval...
$endgroup$
– TheSilverDoe
Mar 12 at 21:02
$begingroup$
@TheSilverDoe you mean the series diverges for all $x$?
$endgroup$
– E.Nole
Mar 12 at 21:06
$begingroup$
First, the function does not depends on $n$. It should be $f$. Secondly, have you determined the definition domain of your function ?
$endgroup$
– TheSilverDoe
Mar 12 at 20:53
$begingroup$
First, the function does not depends on $n$. It should be $f$. Secondly, have you determined the definition domain of your function ?
$endgroup$
– TheSilverDoe
Mar 12 at 20:53
$begingroup$
@TheSilverDoe I was trying to find the domain using Dirichlet's test, the root test and the ratio test but didn't get anything helpful. I initially guessed that the domain is all real $x$ except $0$, but I can't back my argument yet.
$endgroup$
– E.Nole
Mar 12 at 20:58
$begingroup$
@TheSilverDoe I was trying to find the domain using Dirichlet's test, the root test and the ratio test but didn't get anything helpful. I initially guessed that the domain is all real $x$ except $0$, but I can't back my argument yet.
$endgroup$
– E.Nole
Mar 12 at 20:58
$begingroup$
Unfortunately, I think that $f$ is not defined on any interval...
$endgroup$
– TheSilverDoe
Mar 12 at 21:02
$begingroup$
Unfortunately, I think that $f$ is not defined on any interval...
$endgroup$
– TheSilverDoe
Mar 12 at 21:02
$begingroup$
@TheSilverDoe you mean the series diverges for all $x$?
$endgroup$
– E.Nole
Mar 12 at 21:06
$begingroup$
@TheSilverDoe you mean the series diverges for all $x$?
$endgroup$
– E.Nole
Mar 12 at 21:06
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$$sum_n=1^infty fracsin^2(nx)sqrtn = sum_n=1^infty frac1 - cos(2nx)2sqrtn$$
Of course, $sum_n 1/sqrtn$ diverges, while $sum_n cos(2nx)/sqrtn$ converges by Dirichlet's test whenever $sin(x) ne 0$. Therefore your series must diverge whenever $sin(x) ne 0$.
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
$endgroup$
1
$begingroup$
Second series needs a $2$ in the denominator.
$endgroup$
– zhw.
Mar 12 at 21:48
$begingroup$
Oops, yes. Editing.
$endgroup$
– Robert Israel
Mar 13 at 1:46
add a comment |
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1 Answer
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1 Answer
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oldest
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oldest
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active
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votes
$begingroup$
$$sum_n=1^infty fracsin^2(nx)sqrtn = sum_n=1^infty frac1 - cos(2nx)2sqrtn$$
Of course, $sum_n 1/sqrtn$ diverges, while $sum_n cos(2nx)/sqrtn$ converges by Dirichlet's test whenever $sin(x) ne 0$. Therefore your series must diverge whenever $sin(x) ne 0$.
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
$endgroup$
1
$begingroup$
Second series needs a $2$ in the denominator.
$endgroup$
– zhw.
Mar 12 at 21:48
$begingroup$
Oops, yes. Editing.
$endgroup$
– Robert Israel
Mar 13 at 1:46
add a comment |
$begingroup$
$$sum_n=1^infty fracsin^2(nx)sqrtn = sum_n=1^infty frac1 - cos(2nx)2sqrtn$$
Of course, $sum_n 1/sqrtn$ diverges, while $sum_n cos(2nx)/sqrtn$ converges by Dirichlet's test whenever $sin(x) ne 0$. Therefore your series must diverge whenever $sin(x) ne 0$.
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
$endgroup$
1
$begingroup$
Second series needs a $2$ in the denominator.
$endgroup$
– zhw.
Mar 12 at 21:48
$begingroup$
Oops, yes. Editing.
$endgroup$
– Robert Israel
Mar 13 at 1:46
add a comment |
$begingroup$
$$sum_n=1^infty fracsin^2(nx)sqrtn = sum_n=1^infty frac1 - cos(2nx)2sqrtn$$
Of course, $sum_n 1/sqrtn$ diverges, while $sum_n cos(2nx)/sqrtn$ converges by Dirichlet's test whenever $sin(x) ne 0$. Therefore your series must diverge whenever $sin(x) ne 0$.
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
$endgroup$
$$sum_n=1^infty fracsin^2(nx)sqrtn = sum_n=1^infty frac1 - cos(2nx)2sqrtn$$
Of course, $sum_n 1/sqrtn$ diverges, while $sum_n cos(2nx)/sqrtn$ converges by Dirichlet's test whenever $sin(x) ne 0$. Therefore your series must diverge whenever $sin(x) ne 0$.
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
edited Mar 13 at 1:46
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
answered Mar 12 at 21:27
Robert IsraelRobert Israel
327k23216469
327k23216469
1
$begingroup$
Second series needs a $2$ in the denominator.
$endgroup$
– zhw.
Mar 12 at 21:48
$begingroup$
Oops, yes. Editing.
$endgroup$
– Robert Israel
Mar 13 at 1:46
add a comment |
1
$begingroup$
Second series needs a $2$ in the denominator.
$endgroup$
– zhw.
Mar 12 at 21:48
$begingroup$
Oops, yes. Editing.
$endgroup$
– Robert Israel
Mar 13 at 1:46
1
1
$begingroup$
Second series needs a $2$ in the denominator.
$endgroup$
– zhw.
Mar 12 at 21:48
$begingroup$
Second series needs a $2$ in the denominator.
$endgroup$
– zhw.
Mar 12 at 21:48
$begingroup$
Oops, yes. Editing.
$endgroup$
– Robert Israel
Mar 13 at 1:46
$begingroup$
Oops, yes. Editing.
$endgroup$
– Robert Israel
Mar 13 at 1:46
add a comment |
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$begingroup$
First, the function does not depends on $n$. It should be $f$. Secondly, have you determined the definition domain of your function ?
$endgroup$
– TheSilverDoe
Mar 12 at 20:53
$begingroup$
@TheSilverDoe I was trying to find the domain using Dirichlet's test, the root test and the ratio test but didn't get anything helpful. I initially guessed that the domain is all real $x$ except $0$, but I can't back my argument yet.
$endgroup$
– E.Nole
Mar 12 at 20:58
$begingroup$
Unfortunately, I think that $f$ is not defined on any interval...
$endgroup$
– TheSilverDoe
Mar 12 at 21:02
$begingroup$
@TheSilverDoe you mean the series diverges for all $x$?
$endgroup$
– E.Nole
Mar 12 at 21:06