convergence of series

Watch
wilson dang
Badges: 7
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#1
Report Thread starter 5 years ago
#1
ImageImageFrom 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
0
reply
hassassin04
Badges: 17
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#2
Report 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.
0
reply
wilson dang
Badges: 7
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#3
Report Thread starter 5 years ago
#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......
0
reply
hassassin04
Badges: 17
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#4
Report 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.
1
reply
wilson dang
Badges: 7
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#5
Report Thread starter 5 years ago
#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
reply
hassassin04
Badges: 17
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#6
Report 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..
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

When did you submit your Ucas application if you applied to go to university this year?

September 2021 (34)
7.26%
October 2021 (212)
45.3%
November 2021 (52)
11.11%
December 2021 (61)
13.03%
January 2021 (53)
11.32%
I still haven't submitted it yet! (41)
8.76%
Something else (15)
3.21%

Watched Threads

View All
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise