See here for using Latex typesetting.
Use Cauchy Condensation Test. I couldn't find a better way to do it, but there must be..
(Notice that the series is decreasing after n=10)
This sequence diverges, so the series diverge too.
I have four calculus problems that I haven't been able to figure out. The general question for them is "Test the series for convergence or divergence by explicitly showing the test being used."
I've tried the Limit comparison test, Integral test, Ratio test, and Root test. All failed for me. Here's the questions: (all are series from n=1 to infinity, i didn't know how to use this forum's math type)
1. 10^n/n!
2. sin(n)
3. (2n)^n/n^(2n)
4. n^2/e^((n)^3)
Any and all help is greatly appreciated!
See here for using Latex typesetting.
Use Cauchy Condensation Test. I couldn't find a better way to do it, but there must be..
(Notice that the series is decreasing after n=10)
This sequence diverges, so the series diverge too.
I hope that your professor is most concerned that you know and understand the basic ideas here. It seems that you have a great deal of study ahead of you if you are to meet that goal. Thus, if I were you I would not be so concerned with presentation, but rather exhibiting understanding of how series work.
As you can see first example for small n is growing but for higher n factorial faster almost than any useful function except Bessel asimptotic function.
If you have used D'Alambert:
Also you can eveluate with Stirling's formula:
and than you can do anything you want
That sum does not converge; converge only for .
For third example:
Use Cauchy's (roots test) and at the and you got
and goes to 0 when n rise to infinity.
And the last one:
Use comparison test and rough estimation (it is only estimation)
Why you can use this estimation?
Becuse you know that
and also
I try my best to make clear and
Good luck with professor.
Miloš