absolute convergence

[Sarahisme]

hey guys

 

could i get a hint at how to do this problem please? i just can't quite work out what my first step should be,....

 

Thanks

 

-Sarah

[Sarahisme]

how about this....

 

use the limit comparison test...

 

so let |a(n)^{2}| = c(n)

 

lim,n->infinity, (c(n))/(a(n)) = +infinity

 

well ok....maybe not, but at least i'm trying right? :S

[Sarahisme]

ok here is another proof of mine for this question.... i am a little unsure about the inequality bit because well...i sorta made that up somewhat...anyways here it is.....:

[DQW]

The inequality holds; though it should be "strictly less than", unless all the terms are 0. But you can not just "make it up" and hope it's true, so understand why it is true.

[Sarahisme]

well when i say "made it up" i ment i sorta tryed a few cases, and saw that it worked, so then assumed it worked for all cases (quite dodgey i know).

 

would you be able to point me to somewhere where i can find this inequality written out properly (i.e. a maths webpage or something?)

 

also i am gathering if that ineqaulity holds, then the rest of the proof is correct?

 

Thanks for the reply DQW:)

[DQW]

would you be able to point me to somewhere where i can find this inequality written out properly (i.e. a maths webpage or something?)

I doubt you'd find it written out anywhere. All it takes to prove it, is writing out explicitly, each term of the expression on either side of the inequality and multiplying out the terms on the RHS.

[math]eg:~~\sum_n |a_n| = (|a_1|~+~|a_2|~+~|a_3|~+~\dots ) [/math]

[Sarahisme]

oh ok i see, well thanks again DQW

 

yay! i actually got a proof right! (well 90% right anyways )

[Sarahisme]

oh wait well i got the proof right but i still don't know what an example would be...hmmm

[Sarahisme]

The inequality holds; though it should be "strictly less than", unless all the terms are 0. But you can not just "make it up" and hope it's true, so understand why it is true.

 

i am not sure this is true... (i'm probably wrong, so correct if i am )

 

well what if the sum just goes from n = 1 to 1 ??

 

then it would be <=

 

wouldn't it? or ....not?

[DQW]

If the "sum" contained just one term you would have a strict equality [imath]|a_1*a_1| = |a_1|*|a_1| [/imath], but you are told that the series have infinite terms.

[Sarahisme]

hmmm ok, your right (yet again! ), would you be able to give me some help with the second part of the question (...finding the example)

[Sarahisme]

oh i think i've got one.... how bout this series for b(n):

 

...series or should i say sequence...i am not sure

[DQW]

That'll do just fine. And it's a series.

 

Sequence : a1, a2, a3, ...

 

Series : a1 + a2 + a3 + ...

[Dave]

Personally I was going to suggest the alternating harmonic series (saves writing the square root), but that'll do just fine

[Sarahisme]

ok thanks for that guys