Does $sum_{n=1}^{infty} frac{3+(-1)^n}{n}$ converge or diverge?
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I'm having trouble figuring out if the following series converges or diverges.
$$sum_{n=1}^{infty} frac{3+(-1)^n}{n}$$
Here's my thinking:
$$frac{2}{n} leq frac{3+(-1)^n}{n}$$
Since $sum_{n=1}^{infty} frac{2}{n}$ diverges, then so does $sum_{n=1}^{infty} frac{3+(-1)^n}{n}$
Is that correct?
calculus sequences-and-series divergent-series
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
$endgroup$
add a comment |
$begingroup$
I'm having trouble figuring out if the following series converges or diverges.
$$sum_{n=1}^{infty} frac{3+(-1)^n}{n}$$
Here's my thinking:
$$frac{2}{n} leq frac{3+(-1)^n}{n}$$
Since $sum_{n=1}^{infty} frac{2}{n}$ diverges, then so does $sum_{n=1}^{infty} frac{3+(-1)^n}{n}$
Is that correct?
calculus sequences-and-series divergent-series
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
$endgroup$
9
$begingroup$
Yes. correct. continue.
$endgroup$
– hamam_Abdallah
Dec 10 '18 at 20:47
add a comment |
$begingroup$
I'm having trouble figuring out if the following series converges or diverges.
$$sum_{n=1}^{infty} frac{3+(-1)^n}{n}$$
Here's my thinking:
$$frac{2}{n} leq frac{3+(-1)^n}{n}$$
Since $sum_{n=1}^{infty} frac{2}{n}$ diverges, then so does $sum_{n=1}^{infty} frac{3+(-1)^n}{n}$
Is that correct?
calculus sequences-and-series divergent-series
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
$endgroup$
I'm having trouble figuring out if the following series converges or diverges.
$$sum_{n=1}^{infty} frac{3+(-1)^n}{n}$$
Here's my thinking:
$$frac{2}{n} leq frac{3+(-1)^n}{n}$$
Since $sum_{n=1}^{infty} frac{2}{n}$ diverges, then so does $sum_{n=1}^{infty} frac{3+(-1)^n}{n}$
Is that correct?
calculus sequences-and-series divergent-series
calculus sequences-and-series divergent-series
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
edited Dec 22 '18 at 8:37
choco_addicted
8,08261947
8,08261947
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
asked Dec 10 '18 at 20:46
James MitchellJames Mitchell
25627
25627
9
$begingroup$
Yes. correct. continue.
$endgroup$
– hamam_Abdallah
Dec 10 '18 at 20:47
add a comment |
9
$begingroup$
Yes. correct. continue.
$endgroup$
– hamam_Abdallah
Dec 10 '18 at 20:47
9
9
$begingroup$
Yes. correct. continue.
$endgroup$
– hamam_Abdallah
Dec 10 '18 at 20:47
$begingroup$
Yes. correct. continue.
$endgroup$
– hamam_Abdallah
Dec 10 '18 at 20:47
add a comment |
2 Answers
2
active
oldest
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$begingroup$
Note that
$$frac{2}{n}leqfrac{3+(-1)^n}{n},$$
so by the comparison criteria, your series diverges.
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
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add a comment |
$begingroup$
Yes your prove is perfectly fine, indeed note, as an alternative
$$sum_{n=1}^{N} frac{3+(-1)^n}{n}=sum_{n=1}^{N} frac{3}{n}+sum_{n=1}^{N} frac{(-1)^n}{n}$$
and taking the limit $Nto infty$ the first series on the RHS diverges whereas the second one converges (by Leibniz).
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
$endgroup$
2
$begingroup$
Please don't casually split up sums like this without justification. (This one is OK but you have to show why.)
$endgroup$
– Ethan Bolker
Dec 10 '18 at 21:02
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@EthanBolker Yes you are right! I fix, Thanks
$endgroup$
– gimusi
Dec 10 '18 at 21:03
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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$begingroup$
Note that
$$frac{2}{n}leqfrac{3+(-1)^n}{n},$$
so by the comparison criteria, your series diverges.
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
$endgroup$
add a comment |
$begingroup$
Note that
$$frac{2}{n}leqfrac{3+(-1)^n}{n},$$
so by the comparison criteria, your series diverges.
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
$endgroup$
add a comment |
$begingroup$
Note that
$$frac{2}{n}leqfrac{3+(-1)^n}{n},$$
so by the comparison criteria, your series diverges.
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
$endgroup$
Note that
$$frac{2}{n}leqfrac{3+(-1)^n}{n},$$
so by the comparison criteria, your series diverges.
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
answered Dec 10 '18 at 20:58
José Alejandro Aburto AranedaJosé Alejandro Aburto Araneda
802110
802110
add a comment |
add a comment |
$begingroup$
Yes your prove is perfectly fine, indeed note, as an alternative
$$sum_{n=1}^{N} frac{3+(-1)^n}{n}=sum_{n=1}^{N} frac{3}{n}+sum_{n=1}^{N} frac{(-1)^n}{n}$$
and taking the limit $Nto infty$ the first series on the RHS diverges whereas the second one converges (by Leibniz).
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
$endgroup$
2
$begingroup$
Please don't casually split up sums like this without justification. (This one is OK but you have to show why.)
$endgroup$
– Ethan Bolker
Dec 10 '18 at 21:02
$begingroup$
@EthanBolker Yes you are right! I fix, Thanks
$endgroup$
– gimusi
Dec 10 '18 at 21:03
add a comment |
$begingroup$
Yes your prove is perfectly fine, indeed note, as an alternative
$$sum_{n=1}^{N} frac{3+(-1)^n}{n}=sum_{n=1}^{N} frac{3}{n}+sum_{n=1}^{N} frac{(-1)^n}{n}$$
and taking the limit $Nto infty$ the first series on the RHS diverges whereas the second one converges (by Leibniz).
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
$endgroup$
2
$begingroup$
Please don't casually split up sums like this without justification. (This one is OK but you have to show why.)
$endgroup$
– Ethan Bolker
Dec 10 '18 at 21:02
$begingroup$
@EthanBolker Yes you are right! I fix, Thanks
$endgroup$
– gimusi
Dec 10 '18 at 21:03
add a comment |
$begingroup$
Yes your prove is perfectly fine, indeed note, as an alternative
$$sum_{n=1}^{N} frac{3+(-1)^n}{n}=sum_{n=1}^{N} frac{3}{n}+sum_{n=1}^{N} frac{(-1)^n}{n}$$
and taking the limit $Nto infty$ the first series on the RHS diverges whereas the second one converges (by Leibniz).
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
$endgroup$
Yes your prove is perfectly fine, indeed note, as an alternative
$$sum_{n=1}^{N} frac{3+(-1)^n}{n}=sum_{n=1}^{N} frac{3}{n}+sum_{n=1}^{N} frac{(-1)^n}{n}$$
and taking the limit $Nto infty$ the first series on the RHS diverges whereas the second one converges (by Leibniz).
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
edited Dec 10 '18 at 21:04
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
answered Dec 10 '18 at 20:57
gimusigimusi
92.8k84494
92.8k84494
2
$begingroup$
Please don't casually split up sums like this without justification. (This one is OK but you have to show why.)
$endgroup$
– Ethan Bolker
Dec 10 '18 at 21:02
$begingroup$
@EthanBolker Yes you are right! I fix, Thanks
$endgroup$
– gimusi
Dec 10 '18 at 21:03
add a comment |
2
$begingroup$
Please don't casually split up sums like this without justification. (This one is OK but you have to show why.)
$endgroup$
– Ethan Bolker
Dec 10 '18 at 21:02
$begingroup$
@EthanBolker Yes you are right! I fix, Thanks
$endgroup$
– gimusi
Dec 10 '18 at 21:03
2
2
$begingroup$
Please don't casually split up sums like this without justification. (This one is OK but you have to show why.)
$endgroup$
– Ethan Bolker
Dec 10 '18 at 21:02
$begingroup$
Please don't casually split up sums like this without justification. (This one is OK but you have to show why.)
$endgroup$
– Ethan Bolker
Dec 10 '18 at 21:02
$begingroup$
@EthanBolker Yes you are right! I fix, Thanks
$endgroup$
– gimusi
Dec 10 '18 at 21:03
$begingroup$
@EthanBolker Yes you are right! I fix, Thanks
$endgroup$
– gimusi
Dec 10 '18 at 21:03
add a comment |
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$begingroup$
Yes. correct. continue.
$endgroup$
– hamam_Abdallah
Dec 10 '18 at 20:47