Question

Help with Divergence Test: Use the Divergence test to determine whether the following series diverges or state that the test is inconclusive:

Answer 1

\[\sum_{k=2}^{\infty}\frac{ \sqrt{k} }{ \ln ^{10} k}\]

Answer 2

what do you recall about this method?

Answer 3

Well, if a_k goes to 0 as k goes to infinity then the divergence test is inconclusive, but if it goes to any other constant or infinity the series diverges

Answer 4

and ak is the rule that the summation is working with

Answer 5

I am having trouble working with a_k anyalitcally to show it diverges or that the test is inconclusive

Answer 6

how do we determine the limit of a sequence?

Answer 7

in this case, we go ahead and use sqrt(k)/ln^10 k

Answer 8

im thinking a ratio test with that

Answer 9

\[\lim~~\frac{ (k+1)^{1/2}~ln^{10}(k) }{k^{1/2} \ln ^{10}(k+1)}\]
but i dont thik a ratio test will be much use .. at least i cant see a way to work it using that

Answer 10

what other analytic tools are at our disposal? comparison?

Answer 11

i mean this is specifically directed to use the divergenc test

Answer 12

if memory serves; the divergence test looks at the sequence ... so working some test on the sequence to find its limit or lack of a limit is what is required

Answer 13

if ak does not limit to zero .... the by the divergence test yada yada yada

what is a divergence test on a sequence?

Answer 14

do you see the conundrum?

the divergence test of a series rests upon finding the limit of the sequence that it is summing.

the limit of the sequence is determinable by a variety of methods ... none of which are called a divergence test

Answer 15

We are not looking at the sum of a sequence in this case, the divegrence test does not deal with partial summs. We are asked to look at the sequence of numbers produced by ak, in this case sqrt(k)/ln^10 k. I was trying to see if there is an anylitical way to evalute this series