Question

Integral test

Answer 1

\[\Sigma \frac{ 1 }{ (\ln(7n) )^{n}}\]

Answer 2

n=1 to infinity

Answer 3

you are suppose to use the integral test ?

Answer 4

yes I know that it is continuous positive and decreasing
I just cant figure out how to integrate it

Answer 5

I would probably try to use root test
I don't think integral test would work

Answer 6

just because I don't see how to integrate that :p

Answer 7

haven't learned that yet

Answer 8

ratio test?

Answer 9

nope so I cant use it, he will take off points.

Answer 10

so it like totally has to be the integral test?
we can't compare it another similar sum or something?

Answer 11

nope we cant which sucks. how the heck do you integrate this crazy thing

Answer 12

I'm going to ask one more stupid question

Answer 13

ok

Answer 14

\[\frac{1}{(\ln(7n))^n} \text{ or } \frac{1}{\ln(7n^n)} \text{ or } \frac{1}{\ln((7n)^n)}\]

Answer 15

is n a power inside the log( ) thing

Answer 16

Integral is the first test you learn, which is unfortunate haha.
\[f(x) = \frac{ 1 }{ (\ln(7x))^x }\]
Mhm.

Answer 17

if it is that latter one I can help you with that I believe

Answer 18

its the first one you wrote and the one he wrote as well

Answer 19

err

Answer 20

wolfram can't do it either

Answer 21

I kinda wonder if he meant this \[\int\limits_{1}^{\infty}\frac{1}{\ln((7n)^n)} dn \text{ \because this totally doable } \\ \int\limits_{1}^{\infty}\frac{1}{n \ln(7n)} dn \\ \text{ \let } u=\ln(7n) \\ du=\frac{7}{7n} dn=\frac{1}{n} dn \\ \int\limits_1^\infty \frac{1}{n} \cdot \frac{1}{\ln(7n)} dn \\ \int\limits_{\ln(7)}^{\infty} \frac{1}{u} du = \lim_{z \rightarrow \infty}(\ln(z)-\ln(\ln(7)))\]
but we can't do this since we have the n on the whole thingy :(

Answer 22

\[\int\limits_{1}^{\infty} \frac{ 1 }{ (\ln(7x))^x }dx \implies \lim_{t \rightarrow \infty} \int\limits_{1}^{t} \frac{ 1 }{ (\ln(7x))^x }dx\] heh maybe let u = ln(7x)

Answer 23

Nope, won't work

Answer 24

the bottom one has the right set up but I don't know how to finish it, is there a way to integrate 1/a^x

Answer 25

Well \[\int\limits \frac{ 1 }{ a^x }dx = - \frac{ a^{-x} }{ lna }+C\] are you sure we have to use integral test? :O

Answer 26

yes

Answer 27

Haha, I think I'll go think about this for a while and leave it to myininaya, as I'm too dumb to do this.

Answer 28

No you are not too dumb

Answer 29

I think this particular one is impossible for integral test

Answer 30

unless the professor meant something else

Answer 31

well how can we use a comparison test either direct or limit. sorry I click away my computer is stupid.

Answer 32

I think it might be one of those questions where they trick you, and want you to understand it's not possible with the integral test. Such as if you were given \[\sum_{n=1}^{\infty} \ln \left( \frac{ n }{ 2n+5 } \right)\] you know this one isn't possible with the integral test.

Answer 33

?? comparison test

Answer 34

well \[\frac{1}{(\ln(7n))^n} >0 \text{ on } n \in [1,\infty) \]
so let's try to think of a larger function than
\[\frac{1}{(\ln(7n))^n}\]
that maybe we could apply the integral test to

Answer 35

I don't know

Answer 36

http://www.wolframalpha.com/input/?i=y%3D1%2F%28ln%287x%29%29%5Ex%2C+y%3D1%2Fx%5E2%2Cx+is+between+1+and+200
it looks 1/x^2 is bigger that function

Answer 37

\[\frac{1}{n^2}> \frac{1}{(\ln(7n))^n} \\ \text{ so you can \not figure out if this converges } \sum_{n=1}^\infty \frac{1}{n^2}\]
and if this converges then the other converges

Answer 38

Nice :)!