/sci/ - Science & Math

>>9312239
>interval of convergence

???? this isn't a power series

both are trivially divergent since the terms are divergent

>>9312259
Yeah these are the two series yielded from finding the apparent interval of convergence from a power series. What does that mean "trivially divergent since the terms are divergent." How would one prove the divergence of these.

>>9312265
if a series with terms a_n converges, then lim a_n -> 0. use the contrapositive

>>9312271
That's what I'm stuck on though. I'm not sure how to evaluate these limits.

>>9312274
[math]2^n = e^{n ln(2)} > e^n [/math]

which grows faster than any polynomial, which themselves grow faster than [math]\sqrt{n}[/math]

>>9312274
Know what kind of functions grow faster than others. Pic related is helpful.

>>9312277
>>9312281
Okay that all makes a lot of sense.

I'm in a Calc II class though, so I need to be able to mathematically work these out. Can L'Hospital's rule apply to these to show the limit is not zero, and if so, how would it be done?

>>9312239
Ratio test that shit.

>>9312298
That was my thought. I was getting limits not equal to zero but was skeptical of my own work.

Just out of curiosity, are you testing at the ends of the interval of convergence?

>>9312307
Yes. The x term in the power series was [math](x-3)^n[/math] with the endpoints of the interval being 1 and 2.

>>9312281
which category does x^x belong to?

>>9312306
Ratio test, after simplifying will give you a 2 in the numerator, the root will go to 1 (you can move the limit inside since shits continuous and L'Hospital's) 2>1 so both diverge.

>>9312314
Larger than x!

>>9312312
I think you screwed up somewhere, if that's the case. You are supposed to get series where you can't use the ratio test.

>>9312319
therefore exponential > factorial
no?

>>9312327
Is that so...

I redid my work and found that the interval is fucked. Should be [math]5/2 < x < 7/2[/math]. And now, the ratio test can't be used. They're much simpler.

Now looking at pic related, it's a simple comparison test, no?

>>9312277
>[math]2^n > e^n[/math]
>[math]2 > e[/math]
Not even engineering has a theorem for this...

>>9312352
That's more like it. And yes, comparison test does the trick there.

>>9312352
Actually, it's using the limit comp test. I compared both to [math]n^(-1/2)[/math] and the limits exist, therefore both divergent. Easy money.

>>9312360
Nigger how the fuck did you get the second from the first.

>>9312368
n-th root.

>>9312347

[math]n! \approx \sqrt{2n\pi}(\frac{n}{e})^n < n^n [/math]

[math]x^x[/math] doesn't really count as an "exponential" in the context of that chart, so using it to say "exponential > factorial" isn't quite right.

>>9312392
tfw i still can't understand the 'choose' notation after reading the wikipedia page 100 times
sucks to be a brainlet

>>9312399
That's an "n over e" not "n choose e"

>>9312399
>>9312405

>>9312277
[math] e^nln2 = e^n [/math] brainlet. Otherwise >>9312360.

>>9312426
That's clearly false. [math]ln2[/math] is not 1. Also, [math]2^n \equiv e^{nln2} [/math]

>>9312399
It really helps to get used to counting with factorials (!) and then explaining yourself how combinations work.
Step 1) Explain yourself how to count amount of different ways to choose the numbers 1,2,3,...,10. Now generalize this concept to n.
[math]n! [/math]
Step 2) Explain yourself how to count amount of different ways to choose 7 numbers the numbers 1,2,3,...,10. Now generalize this concept to k & n.
[math]\frac{n!}{(n-k)!} [/math]
Step 3) Explain yourself how to count amount of different ways to choose 7 numbers from the numbers 1,2,3,...,10 without considering the order (hint: apply step 1 on the 7 chosen numbers). Now generalize this concept to k & n.
[math]\frac{n!}{k!(n-k)!} [/math]
Boom, you have your own home-cooked binomial bullshit.
Expand your knowledge by generalizing it with choosing two,three,or m colored balls or letters or whatever.

>>9312440
it was fucked formatting m8 the ln didn't stay in the exponent

>>9312479
ln(2) is less than 1, it's still clearly wrong.

>>9312360
true, sorry I'm a faggot I haven't done any math in years

>>9312291
>I need to be able to mathematically work these out
you have a fundamentally terrible approach to these problems. you're trying to just throw recipes at them for them to work. that's not how it goes. you need to use >>9312281 and comparison arguments.

>>9312466
I wish everyone was as cool as you anon