Question

Limit test for series?

Answer 1

On this question I have \[\sum_{n=1}^{\infty} \frac{6n}{n+9}\]

Answer 2

no chance

Answer 3

when I look at what I did before I see that I just did like what we would do before doing L'Hospitals rule and divide out by the highest "n"

so (6/n)/((n/n) + (9/n))?

Answer 4

no chance wut? :p

Answer 5

the degree of the denominator must be greater than the degree of the numerator by more than one

Answer 6

okay I see I got this q wrong, good because I was konfused hahaha.

Answer 7

to do what sir?

Answer 8

in this case the degrees are the same. in fact the limit as \(n\to \infty=6\) not 0 which is necessary (but not sufficient) for convergence

Answer 9

okay I see.. So there is a limit test then?

Answer 10

if in the limit the individual terms do not go to zero, then there is no way for the infinite sum to converge

Answer 11

mhm

Answer 12

you are trying to add an infinite number of numbers , each of which gets closer and closer to 6, there is no way for this to be finite

Answer 13

nth term test?

Answer 14

i have never heard of such a thing, but at the very least if you want any hope of convergence for an infinite sum, the terms must go to zero

Answer 15

set n = infinity and see where it goes? if it's 0 it converges, if it's anything else it diverges?

Answer 16

oh no!

Answer 17

Thyat's what my book calls it :p.

Answer 18

if the terms do not go to zero, then it does not converge
the converse is false
famous example
\[\sum\frac{1}{n}\] the harmonic series diverges, even though the terms go to 0 in the limit

Answer 19

mhm

Answer 20

I understand.

Answer 21

I'm still confused about this limit test business... does it exist?

Answer 22

so the terms must go to zero, but also they must go to zero fast enough
however in your example \(\lim_{n\to\infty}\frac{6n}{n+p}=6\) so the sum does not converge

Answer 23

yeah that' sthe nth term test :p.

Answer 24

as my book calls it..

Answer 25

there are many tests for convergence
comparison test
ratio test
root test
integral test
cauchy "condensation" test
etc

Answer 26

The ones I've learned are nth term test, geo series test, ratio, integral, power, and then the taylor/maclauren.

Answer 27

it is a rather large topic

Answer 28

alternating

Answer 29

yeah that too

Answer 30

I'm trying to bone up on the rest of it for myfinal tomorrow.... That's why I was soo konfused about this limit series... but it doesn't exist LOL.

Answer 31

Weird thing is I got the same answer when I did it the other way.... (6n/n)/((n/n) + (9/n))? = 6/(1+(9/n(infinity))) ===== 6/(1+ 0) = 6?.... @satellite73