Question

Converge or Diverge?

Answer 1

Wolfram says it doesn't converge but I dont know why

Answer 2

I would try the ratio test
unfortunately I have to go, so I can't actually try it
sorry

Answer 3

Ratio test is inconclusive

Answer 4

do the divergence test

Answer 5

Can you explain what is to me? I don't think I remember it

Answer 6

what is this limit

\[\lim_{n\to\infty}\frac{n}{\sqrt{n^2+1}}\]

Answer 7

1?

Answer 8

yes....which is not zero....so....

Answer 9

So it diverges?

Answer 10

yes

Answer 11

comparison test

Answer 12

Can you explain that one too?

Answer 13

find a constant that is larger than \(\tan^{-1}(n)\) for all \(n\in\mathbb{N}\)

Answer 14

Well doesn't tan^-1 go to from -infi to infi?

Answer 15

that is the domain...but the range is bounded

Answer 16

Oh yeah from o to pi?

Answer 17

no

Answer 18

What is it bounded from?

Answer 19

\[-\frac{\pi}{2}<\tan^{-1}(x)<\frac{\pi}{2}\]

Answer 20

the top toggles from - to positive so we know the only way it would converge is if the bottom went to zero, but the bottom will grow slower than the top...this is how I look at it

Answer 21

Yeah @zzr0ck3r I know you can do that for MC questions but for free responses I think we need to do the comparison test @Zarkon is talking about

Answer 22

So what would I do with those things @Zarkon ?

Answer 23

squeeze thrm

Answer 24

well for \(n\in\mathbb{N}\) we have

\[0<\tan^{-1}(n)<\frac{\pi}{2}\]

Answer 25

so

\[\frac{\tan^{-1}(n)}{n^4}<\frac{\pi/2}{n^4}\]

Answer 26

Doesnt pi/2/ n^4 converge though?

Answer 27

-pi/2 <= tan^(-1)(x) <= pi/2
-pi/(2n^4) <= tan^(-1)(x)/n^4<= pi/(2n^4)
0<= tan^-1x <= 0

Answer 28

yes it does

Answer 29

thus by the comparison test we get...

Answer 30

Oh so both converge i see

Answer 31

yes

Answer 32

Do you know the squeez theroem? dan?

Answer 33

Would i compare that to something like the harmonic?

Answer 34

integral test

Answer 35

And could you also explain that.. Sorry I am preparing for a test and I dont remember any of these

Answer 36

compute

\[\int\limits_{2}^{\infty}\frac{1}{x\ln(x)}dx\]

Answer 37

it converges iff the integral is finite

Answer 38

lnlnx

Answer 39

then evaluate at the limits of integration

Answer 40

okay so it goes to infinity so it meants it diverges?

Answer 41

alternating series test

Answer 42

And what are you supposed to do for that? It is supposed to be decreasing and going to 0?

Answer 43

alternating...decreasing ... and the limit must be zero

Answer 44

How do you show that something is decreasing? I know it is but how woudl i show it

Answer 45

let \(a_n\) be your sequence. then \(a_n\) is (strictly) decreasing if \(a_{n}>a_{n+1}\)

Answer 46

you could also writer \(a_n\) as \(f(n)=a_n\)


and show that \(f(x)\) is decreasing

Answer 47

oh okay. thanks for all your help.