# convergence of series

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#1
From the given ans , i knew that it's conditionally convergent (by using alternating test) i can understand the working to show that it's conditionally convergent . But , i also wanna show it as not absolutely convergent ....

to prove that the series is not absolutely convergent , we would use ratio test , when the ratio is bigger than 1 , the series is said to be not absolutely convergent however , i gt 1 (there's no conclusion whether the series is absolutely convergent or not )https://en.wikipedia.org/wiki/Ratio_test
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5 years ago
#2
Compare with harmonic series for instance - 1/n. If you want proof for that one then there are tons available.
Absolutely convergent means the sum of the modulus of your sequence is convergent. that is | ( -1)/ n^1/2 | = 1/n^1/2.
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#3
(Original post by hassassin04)
Compare with harmonic series for instance - 1/n. If you want proof for that one then there are tons available.
Absolutely convergent means the sum of the modulus of your sequence is convergent. that is | ( -1)/ n^1/2 | = 1/n^1/2.
How to show that it's not absolutely convergent? I gt ratio =1, which is inconclusive......
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5 years ago
#4
(Original post by wilson dang)
How to show that it's not absolutely convergent? I gt ratio =1, which is inconclusive......
Ratio test does not always give you the answer, in particular when the ratio is 1... So you must use other methods.
Again, absolutely convergent means that the sum of the mod of your sequence must converge. So you should be looking at the sum of 1/sqrtn and determine whether it converges or not. By comparison test sum 1/sqrtn > sum 1/n . Sum 1/n diverges therefore 1/sqrtn diverges. The proof of harmonic series ( 1/n ) divergence is a popular result.. one way is https://www.youtube.com/watch?v=4yyLfrsSXQQ
You can use other methods of course -e.g. integral test.
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#5
(Original post by hassassin04)
Ratio test does not always give you the answer, in particular when the ratio is 1... So you must use other methods.
Again, absolutely convergent means that the sum of the mod of your sequence must converge. So you should be looking at the sum of 1/sqrtn and determine whether it converges or not. By comparison test sum 1/sqrtn > sum 1/n . Sum 1/n diverges therefore 1/sqrtn diverges. The proof of harmonic series ( 1/n ) divergence is a popular result.. one way is https://www.youtube.com/watch?v=4yyLfrsSXQQ
You can use other methods of course -e.g. integral test.
do you mean to show that the series is not absolutely convergent , we would use ratio test first , and after found that it's not working , we would use the p -series test to show that it's not absolutely convergent ?
0
5 years ago
#6
(Original post by wilson dang)
do you mean to show that the series is not absolutely convergent , we would use ratio test first , and after found that it's not working , we would use the p -series test to show that it's not absolutely convergent ?
Do you understand what does it mean for a series to be absolutely convergent? You do not have to use the ratio test..
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