Question

calculus. limits. with natural log

Answer 1

Determine whether the sequence converges or diverges. If it converges, find the limit.

Answer 2

please can you re-write your formula for a_n?

Answer 3

\[a_n = \frac{ 7 (\ln (n)^2) }{ 11 n }\]

Answer 4

Have you decided which test you're gonna use to test for convergence?

Answer 5

i am attempted to be bold and say that because n is in denominator, the whole thing goes to infinity on the bottom. which = zero. but the "ln" throws me off

Answer 6

2 rounds of l'hospital I think it make the limit more clear

Answer 7

i am really bad at the l' hosp thing... i have only done that once or twice

Answer 8

we have:
\[{\lim _{n \to \infty }}{a_n} = {\lim _{n \to \infty }}\frac{{14}}{{11}}\frac{{\ln n}}{n}\]

Answer 9

\[\lim_{n \rightarrow \infty}\frac{7 (\ln(n))^2}{11n}\]

Answer 10

Since both top and bottom go to infinity I would apply l'hosptial

Answer 11

the problem is my professor does not "teach" anything. And i get lost easy, cuz i have noone teaching me how to approach the questions properly

Answer 12

differentiate top
differentiate bottom

Answer 13

ok.. lemme see. hold on

Answer 14

And I assumed you meant (ln(n))^2
not ln(n^2) right?

Answer 15

yes, the whole thing squared, not just the n

Answer 16

k so we cannot just bring down the power

Answer 17

\[\frac{d}{dn}(\ln(n))^2 \text{ for this you need chain rule }\]

Answer 18

\[7 (\ln (n))^2\]

Answer 19

\[\frac{d}{dn}(\ln(n))^2=2(\ln(n))^{2-1} \cdot (\ln(n))'\]

Answer 20

yes we can come back and multiply the constant multiple in

Answer 21

do you know the derivative of ln(x) w.r.t x?

Answer 22

Oh now I see how @Michele_Laino got what he sis.

Answer 23

did*

Answer 24

sorry, i keep getting kicked off the connection to this site. idk why

Answer 25

deriv of lnx = 1/x right?

Answer 26

yah!

Answer 27

ok... see how easy i get confused?! lol

Answer 28

so is this 2/x * 1/x if we used x instead of n up there

Answer 29

oh you have the prime next to the n.... that means n=1, right?

Answer 30

\[\lim_{n \rightarrow \infty} \frac{7 (\ln(n))^2}{11n} =\lim_{n \rightarrow \infty}\frac{(7 (\ln(n))^2)'}{(11n)'} \\ = \lim_{n \rightarrow \infty}\frac{7 \cdot 2(\ln(n))^{2-1} \frac{1}{n}}{11} =\lim_{n \rightarrow \infty}\frac{14 \ln(n)}{11 n}\]
So since again you have infty/infty you can apply l'hosptial again

Answer 31

this will be last round of l'hosptial I promise

Answer 32

1/n ?
f(n)*1/n=f(n)/n

Answer 33

I see her question, 1/n became just n. that got me for a sec too

Answer 34

That's where the 11 multiplied the n after the derivative of n was taken.

Answer 35

dripped to the denominator

Answer 36

dropped lol

Answer 37

I like dripped lol

Answer 38

me too! ill keep it

Answer 39

anyways this round is tons easier than the last round
less chain rule

Answer 40

freckles..lm writing down what you did real quick. I HAVE to learn how to do this

Answer 41

\[\lim_{n \rightarrow \infty} \frac{7 (\ln(n))^2}{11n} =\lim_{n \rightarrow \infty}\frac{(7 (\ln(n))^2)'}{(11n)'} \\ = \lim_{n \rightarrow \infty}\frac{7 \cdot 2(\ln(n))^{2-1} \frac{1}{n}}{11} =\lim_{n \rightarrow \infty}\frac{14 \ln(n)}{11 n} \\ =\lim_{n \rightarrow \infty} \frac{(14 \ln(n))'}{(11n)'}=\lim_{n \rightarrow \infty}\frac{14 \frac{1}{n}}{11}\]
I guess what you can do to that 1/n

Answer 42

dripped the n lol

Answer 43

you only need to recall what happens to 1/n as n approaches infinity

Answer 44

and I think you said this number earlier :)

Answer 45

yes.. i know that 1/n would be zero BUT why did we keep "ln" in there and not put 1/n?

Answer 46

i just see LN still and my brain goes, IDK what to do with this thing

Answer 47

Then, i saw your "SECOND" round

Answer 48

i was writing your first round and did not see you post the second....

Answer 49

i hate trig! lol i am no good at it

Answer 50

i feel like calculus is trig!

Answer 51

calculus isn't trig, it's mostly derivations of complex algebra with trig involved :P

Answer 52

Did you take a trig class before calculus?
I have seen a lot of people make that mistake.
Not take trig before cal.

Answer 53

i did take trig... but it was a yr and half before calc... and i think i forgot or didnt retain

Answer 54

If you're relying on your highschool trig to suffice your knowledge of trig integrations in calculus, you're really in a doozy.

Answer 55

But also I don't see ln(x) as a trig function?
Are you just looking at other problems or something?

Answer 56

no.. its just that, everytime i see ln or e... it seems to be in same line of questions that have cos or sin or tan...

Answer 57

My cats took my chair so I'm trying to be respectful to them a little and not retake my chair.

Answer 58

lol

Answer 59

ln(x) is just its in own category, you learn about logarithmic functions, \(\log_a (x)\), exponential growth, \(e^x\), and logarithmic's sister component, natural logs, \(\ln(x)\)

Answer 60

im sure ill have another problem to post, I have one more assignment to do for this week

Answer 61

high school was 11 yrs ago for me lol

Answer 62

funny functions category:
ln(x)
exp(x)
sin(x)
cos(x)
tan(x)
sec(x)
csc(x)
cot(x)

--- trying to think of another funny function
like I don't consider x^3 a funny function because it is easy to write out what this means
x^3=x*x*x

Answer 63

Like what I call a funny function is what you are calling trig function
right?

Answer 64

yes! i hate ALL of those lol

Answer 65

that is why i get same question wrong over and over... and dont even notice! because i see these "funny" looking funcitons and get all freaked out

Answer 66

lol
by the way this is not a universal term for those functions

there are probably some other functions I consider funny but I can't of them off the top of my head

Answer 67

i understand. I like the way you put it tho. just like "dripping" lol... these weird things help me remember easier

Answer 68

i have been watching a lot of "integral calc" on youtube. that girl has helped me A LOT. and then you freckles, have helped me the most on this site. if it wasnt for these two options, i would be failing this course. I've never had such a bad professor

Answer 69

That's why so many people resort to OS for help and guidance :D

Answer 70

it needs a like button like fb tho lol

Answer 71

Ah how sweet