Question

Find every real number k for which the series converges: sum from n=2 to infinity of 1/(n^k * ln(n))

Answer 1

every k greater then 1

Answer 2

think about it this way, it is a logarithmic * exponential. this cancels it out and allows them to eventually converge

Answer 3

cancels it out? I don't follow

Answer 4

logarithmic function gets smaller and smaller, while exponential gets bigger and bigger. if you divide 1 by a function that gets smaller and smaller, you will get a very big number no? like for example 1/.0000001 = big number.. now if you divide 1 by a big big number, it will eventually become 0! so as long as X is exponential(positive direction) it cancels out the properties of the log function.

Answer 5

So if I wanted to prove that, could I somehow use the comparison test?

Answer 6

I don't really remember my tests, but this is just by basic properties of sequences. Sorry, :(

Answer 7

Maybe if I ask it this way, how could I prove that 1/(n^.5 ln(n)) diverges?