Mixing
Does the series $sum_{n=1}^infty 1/(n+n cos(n))$ converge or diverge?
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Does the series $sum_{n=1}^infty 1/(n+n cos(n))$ converge or diverge? How can I use Direct Comparison Test for this problem?
sequences-and-series limits
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
asked Feb 3 at 2:49
KyleKyle
111
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add a comment |
$begingroup$
Does the series $sum_{n=1}^infty 1/(n+n cos(n))$ converge or diverge? How can I use Direct Comparison Test for this problem?
sequences-and-series limits
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
asked Feb 3 at 2:49
KyleKyle
111
$endgroup$
add a comment |
$begingroup$
Does the series $sum_{n=1}^infty 1/(n+n cos(n))$ converge or diverge? How can I use Direct Comparison Test for this problem?
sequences-and-series limits
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
asked Feb 3 at 2:49
KyleKyle
111
$endgroup$
Does the series $sum_{n=1}^infty 1/(n+n cos(n))$ converge or diverge? How can I use Direct Comparison Test for this problem?
sequences-and-series limits
sequences-and-series limits
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
asked Feb 3 at 2:49
KyleKyle
111
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
asked Feb 3 at 2:49
KyleKyle
111
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
edited Feb 3 at 4:50
J. W. Tanner
4,7871420
4,7871420
asked Feb 3 at 2:49
KyleKyle
111
asked Feb 3 at 2:49
KyleKyle
111
asked Feb 3 at 2:49
KyleKyle
111
111
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Notice that $sum_{n=1}^{infty}frac{1}{n+ncos(n)}≥ sum_{n=1}^{infty}frac{1}{n+n}$ (Why?)
Now:
$sum_{n=1}^{infty}frac{1}{2n}=frac{1}{2}sum_{n=1}^{infty}frac{1}{n}$ diverges (Why?)
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
$endgroup$
add a comment |
$begingroup$
Observe that
$$ n + n cos n leq 2n $$
from which one easily sees the divergence of the series after taking reciprocals. (notice tha Harmonic $sum frac{1}{2n} $ diverge)
answered Feb 3 at 2:56
JamesJames
2,636425
$endgroup$
add a comment |
$begingroup$
$a_n=dfrac{1}{n+n cos{n} }$ and let $b_n=dfrac{1}{n}$
$lim_{n to infty}dfrac{a_n}{b_n}=dfrac{1}{1+cosn}=finite $
$sum_n a_n $ and $sum_n b_n$ converge or diverge together.$sum b_n$ is a divergent or convergent series ?
answered Feb 3 at 3:18
Daman deepDaman deep
756420
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Notice that $sum_{n=1}^{infty}frac{1}{n+ncos(n)}≥ sum_{n=1}^{infty}frac{1}{n+n}$ (Why?)
Now:
$sum_{n=1}^{infty}frac{1}{2n}=frac{1}{2}sum_{n=1}^{infty}frac{1}{n}$ diverges (Why?)
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
$endgroup$
add a comment |
$begingroup$
Notice that $sum_{n=1}^{infty}frac{1}{n+ncos(n)}≥ sum_{n=1}^{infty}frac{1}{n+n}$ (Why?)
Now:
$sum_{n=1}^{infty}frac{1}{2n}=frac{1}{2}sum_{n=1}^{infty}frac{1}{n}$ diverges (Why?)
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
$endgroup$
add a comment |
$begingroup$
Notice that $sum_{n=1}^{infty}frac{1}{n+ncos(n)}≥ sum_{n=1}^{infty}frac{1}{n+n}$ (Why?)
Now:
$sum_{n=1}^{infty}frac{1}{2n}=frac{1}{2}sum_{n=1}^{infty}frac{1}{n}$ diverges (Why?)
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
$endgroup$
Notice that $sum_{n=1}^{infty}frac{1}{n+ncos(n)}≥ sum_{n=1}^{infty}frac{1}{n+n}$ (Why?)
Now:
$sum_{n=1}^{infty}frac{1}{2n}=frac{1}{2}sum_{n=1}^{infty}frac{1}{n}$ diverges (Why?)
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
answered Feb 3 at 2:57
babemcnuggetsbabemcnuggets
116110
116110
add a comment |
add a comment |
$begingroup$
Observe that
$$ n + n cos n leq 2n $$
from which one easily sees the divergence of the series after taking reciprocals. (notice tha Harmonic $sum frac{1}{2n} $ diverge)
answered Feb 3 at 2:56
JamesJames
2,636425
$endgroup$
add a comment |
$begingroup$
Observe that
$$ n + n cos n leq 2n $$
from which one easily sees the divergence of the series after taking reciprocals. (notice tha Harmonic $sum frac{1}{2n} $ diverge)
answered Feb 3 at 2:56
JamesJames
2,636425
$endgroup$
add a comment |
$begingroup$
Observe that
$$ n + n cos n leq 2n $$
from which one easily sees the divergence of the series after taking reciprocals. (notice tha Harmonic $sum frac{1}{2n} $ diverge)
answered Feb 3 at 2:56
JamesJames
2,636425
$endgroup$
Observe that
$$ n + n cos n leq 2n $$
from which one easily sees the divergence of the series after taking reciprocals. (notice tha Harmonic $sum frac{1}{2n} $ diverge)
answered Feb 3 at 2:56
JamesJames
2,636425
answered Feb 3 at 2:56
JamesJames
2,636425
answered Feb 3 at 2:56
JamesJames
2,636425
answered Feb 3 at 2:56
JamesJames
2,636425
2,636425
add a comment |
add a comment |
$begingroup$
$a_n=dfrac{1}{n+n cos{n} }$ and let $b_n=dfrac{1}{n}$
$lim_{n to infty}dfrac{a_n}{b_n}=dfrac{1}{1+cosn}=finite $
$sum_n a_n $ and $sum_n b_n$ converge or diverge together.$sum b_n$ is a divergent or convergent series ?
answered Feb 3 at 3:18
Daman deepDaman deep
756420
$endgroup$
add a comment |
$begingroup$
$a_n=dfrac{1}{n+n cos{n} }$ and let $b_n=dfrac{1}{n}$
$lim_{n to infty}dfrac{a_n}{b_n}=dfrac{1}{1+cosn}=finite $
$sum_n a_n $ and $sum_n b_n$ converge or diverge together.$sum b_n$ is a divergent or convergent series ?
answered Feb 3 at 3:18
Daman deepDaman deep
756420
$endgroup$
add a comment |
$begingroup$
$a_n=dfrac{1}{n+n cos{n} }$ and let $b_n=dfrac{1}{n}$
$lim_{n to infty}dfrac{a_n}{b_n}=dfrac{1}{1+cosn}=finite $
$sum_n a_n $ and $sum_n b_n$ converge or diverge together.$sum b_n$ is a divergent or convergent series ?
answered Feb 3 at 3:18
Daman deepDaman deep
756420
$endgroup$
$a_n=dfrac{1}{n+n cos{n} }$ and let $b_n=dfrac{1}{n}$
$lim_{n to infty}dfrac{a_n}{b_n}=dfrac{1}{1+cosn}=finite $
$sum_n a_n $ and $sum_n b_n$ converge or diverge together.$sum b_n$ is a divergent or convergent series ?
answered Feb 3 at 3:18
Daman deepDaman deep
756420
answered Feb 3 at 3:18
Daman deepDaman deep
756420
answered Feb 3 at 3:18
Daman deepDaman deep
756420
answered Feb 3 at 3:18
Daman deepDaman deep
756420
756420
add a comment |
add a comment |
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