How in the world do you determine the following series' convergence/divergence specifically using the limit comparison test?
from n=1 to n=infinity
ln (2n+1)/ (n^2 + 2n)
I'm bamboozled and befuddled!
I know it's against the rule to bump topics but I don't care, I really need help. I might as well post another topic.
Anyways, I followed Opalag's directions but, strangely, I got zero. According to the limit comparison test, if the number equals zero or infinity, then the result is inconclusive. Please clarify?
Also, why would I not use like n^2 instead of sqrt(n)? If I had done that then the series appears to diverge!
Please help. Thanks a lot!
If , and you choose for some p between and 2, then will converge (because p>1), and will be 0 (because p<2).
I hope that makes it clearer. (If you get an infraction for bumping, tell the Moderator that I think you were justified on this occasion.)
Why must p be between 1 and 2? You simplified the series so as to exclude the ln expression in order to dig up the right-handed series for the b series. In that process of simplification, you can input any number in the bottom left and and the top right, and each number brings a different convergence/divergence result!
Also, you seems to contradict yourself. If an/bn is nonzero and finite and if bn converges, then an converges. But you says an/bn converges to zero!
Let me repeat my understanding for the limit comparison test-
To evaluate the series an, pick bn based on an's highest ordered term in each numerator and denominator. Then structure an/bn and look for the limit. If the limit is between zero and infinity and if bn converges, then an/bn converges. But an/bn converges to zero! I'm so confused.
Please don't make me bump this topic again.
If you take and , then as . therefore, by the limit comparison test in the form that I stated, converges.
My book gives the definition of the limit comparison test in the following -
"Let an and bn be positive-termed series. Let L = lim (n to infinity) an/bn. If 0 < L < +infinity, then an and bn either both converge or both diverge. If L = 0 or L = +infinity, then the test is inconclusive."
The issue here is the domain of the limit comparison test and whether it includes zero. It's there, I copied exactly as my book puts it. What's going on?
I really appreciate the fact that you replied speedily!
In fact, I think that your book is somewhat misleading in claiming that the test is inconclusive when the limit is zero or infinity. As my version of the test shows, it is possible to draw conclusions in these cases. However, in those situations the test can only be used to get an implication in one direction; in these cases it is not an "if and only if" test.