kinda hard question: Which of these series is absolutely convergent?

Question

anonymous · Answer

attachment

anonymous · Answer

the first is the only one that gave a clear cut answer of 2/9

anonymous · Answer

might that be it?

anonymous · Answer

ANYONEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

perl · Answer

absolutely convergent, means convergent with absolute value bars, right?

perl · Answer

conditionally convergent allows the possibility of negative terms (alternating series)

perl · Answer

only one of those series is absolutely convergent,

anonymous · Answer

right

perl · Answer

lets start by eliminating choices. 
whats the problem with the second choice

anonymous · Answer

wolfram alpha gives 2/9 and that seems pretty damn absolute to me

anonymous · Answer

haha

perl · Answer

that is the alternating series, the absolute convergence is different

anonymous · Answer

so could it be a?

perl · Answer

oh youre talking about the first choice

perl · Answer

yes but you inputted it incorrectly, we are interested in the absolute value of terms of the series

perl · Answer

http://www.wolframalpha.com/input/?i=sum+%28n%3D1%2C+infinity%29abs+%28%28-1%29^%28n-1%29+n+%2F+2^n%29

perl · Answer

you see, using your logic, choice b converges too http://www.wolframalpha.com/input/?i=sum+%28n%3D1%2C+infinity%29+%28-1%29^%28n-1%29+n^2+%2F+%28n^3+%2B+1%29+

anonymous · Answer

ohhhhhh so its just convergence but with absolute value?

perl · Answer

right

perl · Answer

here is choice b) with absolute values http://www.wolframalpha.com/input/?i=sum+%28n%3D1%2C+infinity%29+abs%28%28-1%29^%28n-1%29+n^2+%2F+%28n^3+%2B+1%29+%29

perl · Answer

that narrows down the choices to one answer

anonymous · Answer

right, actually my logic was that a gave the neatest answer, a nice and simple 2/9, which i assumed was 'absolute' in that it was totally unambiguous,

anonymous · Answer

lol

perl · Answer

ahh, be careful about terminology

perl · Answer

why wasn't choice b) totally unambiguous?

perl · Answer

using that reasoning

perl · Answer

oh because it didn't give a nice answer ( a fraction) ?

anonymous · Answer

huh so it was b, except now i actually know why

perl · Answer

the answer should be a)

anonymous · Answer

yeah i thought b was too drawn out, lmao i know, not how a mathematician should think

anonymous · Answer

right sorry i meant a

perl · Answer

i see

anonymous · Answer

because it gives a 2 (meaning it converges) with the absolute value on the series, while none of the other options do

anonymous · Answer

right?

perl · Answer

correct

perl · Answer

the other choices are what we call 'conditionally convergent' , they converge if you allow negative terms

perl · Answer

so it turns out that absolute convergence is the stronger condition. stronger in the sense that if you know a series is absolutely convergent, it is automatically conditionally convergent as well, i.e. you don't have to check conditional convergence.

 but the other direction is not true. a series that is conditionally convergent may or may not be absolutely convergent.

perl · Answer

theres a simple proof of this, but it escapes me at the moment

perl · Answer

$$ 
\Large \sum |a_n|~ \rm{converges}\Rightarrow \sum a_n~ converges

$$

anonymous · Answer

damn, it gives me great honor to present you with a medal

perl · Answer

$$ 
\Large \sum a_n~ m{converges}
Rightarrow \sum |a_n|~ converges

$$

perl · Answer

aww, thanks :)

anonymous · Answer

you really are deserving of your reputation perl! i also feel like ive been kissed by adriana grande,

anonymous · Answer

hahah that should be a meme -

perl · Answer

heh

perl · Answer

thanks again :)

perl · Answer

nice name by the way

anonymous · Answer

hey you think you could give a quick hint?

anonymous · Answer

i need to find the 'center of the power series'

anonymous · Answer

know any widget i could use? wolfram doesnt seem to be doing any favors