I can't remember how to do series questions but the integral you state and your working looks okay to me.
Classify the following series as convergent or divergent. Provide justification (eg. which convergence test used.). For those series containing symbol explain clearly for which real values of if any the series converges or diverges.
(a)
(b)
(c)
(d)
And please kindly check if my working below is accurate or if I am on the right track.
The question is to evaluate the improper integral .
Workings :
Integration by Substitution,
Let
At
Integration by Parts,
Let
Allowing
.
Thank you!
Hello,
For a), use the root test
For b), use the alternating series or compare it to the Riemann alternating series ( converges iff )
For c), use the ratio test
For d), you can look here for Bertrand series
Hello, I don't really understand the root test as I didn't learn that in my lectures.
As for part (c),
I got
Am I doing the correct thing, if so, how do I proceed from here?
I have not learnt the Bertrand series before either.
The list of test that I learnt was the divergence (DWMT) test, Comparison test, Limit Comparison test, Alternating Series test and ratio test.
From the above tests I learned, which is applicable? I'll keep trying and try to learn up the tests you suggested, but hopefully I'd be able to apply the tests I learned!
Hope to hear from you soon. Thank you!
Hmm it's just an equivalent of the ratio test :
if then it converges (I admit it is not the first formula the wikipedia gives)
for a), I have another way to propose to you.
You have
Noting that , you can say that for all n.
So
Does this series converge ?
Ooops, I'm sorry ! Actually, I guess it's better to use an alternating series test (I missed the (-1)^n part )As for part (c),
I got
Am I doing the correct thing, if so, how do I proceed from here?
In fact, I put this link so that you can see how they deal with such functions, how they conclude why it converges or diverges. It's similar here.I have not learnt the Bertrand series before either.
Excuse me in advance, but I don't know how to use the integral test, so maybe you can use it for one or two series here. Feel free to tryThe list of test that I learnt was the divergence (DWMT) test, Comparison test, Limit Comparison test, Alternating Series test and ratio test.
From the above tests I learned, which is applicable? I'll keep trying and try to learn up the tests you suggested, but hopefully I'd be able to apply the tests I learned!
Hope to hear from you soon. Thank you!
For the alternating test for (b) is the and we prove that this is convergent/divergent hence conclude by alternating test?
For the (c), I reckon the right and this should be divergent because factorials grow at a much faster rate hence by alternating test this series is divergent?
Hmm I am not too sure actually, and what do they mean by explain clearly for which real values of if any the series converges and for which real values of the series diverges?