Convergence/Divergence of infinite series $sumlimits_{n=1}^{infty} frac{1}{n^{1+left|{cos n}right|}}$
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It is well known that $ displaystylesum_{n=1}^{infty} frac{1}{n}$ is divergent while $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+epsilon}}$ is convergent for a fixed positive value of $epsilon$.
It is not difficult to show that $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+frac{1}{n}}}$ is divergent using Limit comparison test with $ displaystylefrac{1}{n}$. There is a post on this question here.
Now comes my questions:
(i) Is $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+left|{cos n}right|}}$ convergent or divergent? (I have tried several tests, like: comparison/limit comparison tests, but fail to get conclusion. My intuition is that it is divergent...)
(ii) It was stated here that $ displaystylesum_{n=1}^{infty} frac{1}{n^{2-cos n}}=sum_{n=1}^{infty} frac{1}{n^{1+(1-cos n)}}$ is divergent. So is there is general way to determine if $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+f(n)}}$ with $f(n)>0$ for all natural number $n$, a convergent or divergent series?
Any comment or answer?
calculus sequences-and-series
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
asked Feb 22 '13 at 0:57
pipipipi
1,133723
$endgroup$
add a comment |
$begingroup$
It is well known that $ displaystylesum_{n=1}^{infty} frac{1}{n}$ is divergent while $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+epsilon}}$ is convergent for a fixed positive value of $epsilon$.
It is not difficult to show that $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+frac{1}{n}}}$ is divergent using Limit comparison test with $ displaystylefrac{1}{n}$. There is a post on this question here.
Now comes my questions:
(i) Is $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+left|{cos n}right|}}$ convergent or divergent? (I have tried several tests, like: comparison/limit comparison tests, but fail to get conclusion. My intuition is that it is divergent...)
(ii) It was stated here that $ displaystylesum_{n=1}^{infty} frac{1}{n^{2-cos n}}=sum_{n=1}^{infty} frac{1}{n^{1+(1-cos n)}}$ is divergent. So is there is general way to determine if $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+f(n)}}$ with $f(n)>0$ for all natural number $n$, a convergent or divergent series?
Any comment or answer?
calculus sequences-and-series
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
asked Feb 22 '13 at 0:57
pipipipi
1,133723
$endgroup$
1
$begingroup$
If you replace cosine with sine, the answer is here: math.stackexchange.com/questions/270064/…
$endgroup$
– user940
Feb 22 '13 at 1:06
$begingroup$
@ByronSchmuland Thanks! From the link provided, some post mentioned similar questions...
$endgroup$
– pipi
Feb 23 '13 at 2:25
add a comment |
$begingroup$
It is well known that $ displaystylesum_{n=1}^{infty} frac{1}{n}$ is divergent while $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+epsilon}}$ is convergent for a fixed positive value of $epsilon$.
It is not difficult to show that $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+frac{1}{n}}}$ is divergent using Limit comparison test with $ displaystylefrac{1}{n}$. There is a post on this question here.
Now comes my questions:
(i) Is $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+left|{cos n}right|}}$ convergent or divergent? (I have tried several tests, like: comparison/limit comparison tests, but fail to get conclusion. My intuition is that it is divergent...)
(ii) It was stated here that $ displaystylesum_{n=1}^{infty} frac{1}{n^{2-cos n}}=sum_{n=1}^{infty} frac{1}{n^{1+(1-cos n)}}$ is divergent. So is there is general way to determine if $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+f(n)}}$ with $f(n)>0$ for all natural number $n$, a convergent or divergent series?
Any comment or answer?
calculus sequences-and-series
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
asked Feb 22 '13 at 0:57
pipipipi
1,133723
$endgroup$
It is well known that $ displaystylesum_{n=1}^{infty} frac{1}{n}$ is divergent while $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+epsilon}}$ is convergent for a fixed positive value of $epsilon$.
It is not difficult to show that $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+frac{1}{n}}}$ is divergent using Limit comparison test with $ displaystylefrac{1}{n}$. There is a post on this question here.
Now comes my questions:
(i) Is $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+left|{cos n}right|}}$ convergent or divergent? (I have tried several tests, like: comparison/limit comparison tests, but fail to get conclusion. My intuition is that it is divergent...)
(ii) It was stated here that $ displaystylesum_{n=1}^{infty} frac{1}{n^{2-cos n}}=sum_{n=1}^{infty} frac{1}{n^{1+(1-cos n)}}$ is divergent. So is there is general way to determine if $ displaystylesum_{n=1}^{infty} frac{1}{n^{1+f(n)}}$ with $f(n)>0$ for all natural number $n$, a convergent or divergent series?
Any comment or answer?
calculus sequences-and-series
calculus sequences-and-series
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
asked Feb 22 '13 at 0:57
pipipipi
1,133723
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
asked Feb 22 '13 at 0:57
pipipipi
1,133723
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
edited Dec 11 '18 at 7:15
Martin Sleziak
44.8k9118272
44.8k9118272
asked Feb 22 '13 at 0:57
pipipipi
1,133723
asked Feb 22 '13 at 0:57
pipipipi
1,133723
asked Feb 22 '13 at 0:57
pipipipi
1,133723
1,133723
1
$begingroup$
If you replace cosine with sine, the answer is here: math.stackexchange.com/questions/270064/…
$endgroup$
– user940
Feb 22 '13 at 1:06
$begingroup$
@ByronSchmuland Thanks! From the link provided, some post mentioned similar questions...
$endgroup$
– pipi
Feb 23 '13 at 2:25
add a comment |
1
$begingroup$
If you replace cosine with sine, the answer is here: math.stackexchange.com/questions/270064/…
$endgroup$
– user940
Feb 22 '13 at 1:06
$begingroup$
@ByronSchmuland Thanks! From the link provided, some post mentioned similar questions...
$endgroup$
– pipi
Feb 23 '13 at 2:25
1
1
$begingroup$
If you replace cosine with sine, the answer is here: math.stackexchange.com/questions/270064/…
$endgroup$
– user940
Feb 22 '13 at 1:06
$begingroup$
If you replace cosine with sine, the answer is here: math.stackexchange.com/questions/270064/…
$endgroup$
– user940
Feb 22 '13 at 1:06
$begingroup$
@ByronSchmuland Thanks! From the link provided, some post mentioned similar questions...
$endgroup$
– pipi
Feb 23 '13 at 2:25
$begingroup$
@ByronSchmuland Thanks! From the link provided, some post mentioned similar questions...
$endgroup$
– pipi
Feb 23 '13 at 2:25
add a comment |
0
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$begingroup$
If you replace cosine with sine, the answer is here: math.stackexchange.com/questions/270064/…
$endgroup$
– user940
Feb 22 '13 at 1:06
$begingroup$
@ByronSchmuland Thanks! From the link provided, some post mentioned similar questions...
$endgroup$
– pipi
Feb 23 '13 at 2:25