Simple Series / Sequence Questions

[khauna]

I am having a little trouble deducing how to solve general series question.

1 ) how do you determine a series is decreasing?
for example:
\(\displaystyle \sum\limits_{n = 1}^\infty {(lnn/n)^2 }\\)

i know your supposed to do something like \(\displaystyle (ln(n+1)/(n+1))^2<(lnn/n)^2\)
((should be greater than or equal to sign, not sure how to do it))

but i know if you can prove inequalities created like that, then the function is decreasing, but that inequality its too difficult and or time consuming for a test. So then you resort to taking the derivative (as our teacher tell us)
which yields:

\(\displaystyle {(2)(lnn/n)*(1-lnn)/(n^2)}\) i dont see how the derivative tells us if the equation is negative. Is it because the derivative is negative? and do you determine that by plugging in a random number?


Also ill take any advice to figuring out what kind of test to use to determine if a series is convergent or divergent
like: Comparison test, Limit comparison test, integral test, n-th root test, ratio test, alternating series test, absolute convergance test...think thats all of them

like i know problems with factorials, ratio test is the best option, and problems with natural log, integral test is always a good choice...but yea, im having trouble picking the right ones...

 

[DR._Glockman]

First of all when in doubt, expand the summation five or more times to get a grasp upon the problem.

Then take it from there.

 

[pka]

You can fall back on basic caculus
\(\displaystyle f(x) = \frac{{\ln (x)}}{x} \Rightarrow \quad f'(x) = \frac{{1 - \ln (x)}}{{x^2 }}\)
\(\displaystyle \left( {n \ge 3} \right)\left[ {f'(n) < 0} \right]\quad \Rightarrow \quad 0 < f^2 (n + 1) < f^2 (n)\)