Series and Integral Test Question

[ardentmed]

Hey guys,

I have another quick question for the current problem set I'm working on. I'm unsure about a in particular.

for a, does it fulfill the integral test's requirements? It most certainly seems to be positive and decreasing. That can be proven by taking dy/dx of lnx/(x^2), right? I ended up getting (-1/x) * (lnx + 1 [from infinity to 2]), which gives me infinity/infinity. Does that make it divergent?

As for b,c, and d. I got convergent by the limit comparison test, convergent by the ratio test, and convergent by the ratio test, respectively.



Thanks again.

 

[ardentmed]

Hey guys,

I have another quick question for the current problem set I'm working on. I'm unsure about a in particular.

for a, does it fulfill the integral test's requirements? It most certainly seems to be positive and decreasing. That can be proven by taking dy/dx of lnx/(x^2), right? I ended up getting (-1/x) * (lnx + 1 [from infinity to 2]), which gives me infinity/infinity. Does that make it divergent?

As for b,c, and d. I got convergent by the limit comparison test, convergent by the ratio test, and convergent by the ratio test, respectively.



Thanks again.

So can anyone help me out at least with question a?

Thanks again.

 

[lookagain]

Hey guys,

I have another quick question for the current problem set I'm working on. I'm unsure about a in particular.

for a, does it fulfill the integral test's requirements? It most certainly seems to be positive and decreasing.
That can be proven by taking dy/dx of lnx/(x^2), right? I ended up getting (-1/x) * (lnx + 1 [from infinity to 2]),
which gives me infinity/infinity. Does that make it divergent?

As for b,c, and d. I got convergent by the limit comparison test, convergent by the ratio test, and convergent by the ratio test,
respectively.

So, for instance, according to this site:


http://www2.fiu.edu/~wlodarcz/series-integral%20and%20comparison.pdf

f(x) must be positive, continuous, and decreasing on the interval [2, oo).


\(\displaystyle Here, \ \ f'(x) \ = \ \dfrac{1 \ - \ 2\ln(x)}{x^3}\)


Check the interval against the first derivative to see that the original function, f(x),
is always decreasing on the given interval.