I am really poor in this part, been trying to do as much exercises as possible to get familiar with the patterns and I realised there are these few patterns that I couldn't seem to approach it.
Sometimes I know the right approach for it, but I have troubles simplifying the terms in order to prove that it fits certain criteria required to do conclusions. Here are a few cases, I hope that you guys would be able to give me a guidance on how to approach these different cases.
Case I : When there is root of something involved.
(i)
(ii)
(iii)
Case II : When there is to the power of n.
(i)
(ii)
I reckon you approach this one with the ratio test? I always have troubles simplifying them!
Case III : When there is polynomials.
(i)
(ii)
And the following few questions, I don't know how to classify them.
(i) - Alternating Series??
(ii)
(iii) - I suppose you use Ratio test? But I have troubles simplifying this one!
Quite a number of questions there, I know! So sorry for the overload of questions Thanks for looking and helping though!!
Your help to master this topic would be greatly greatly appreciated!
Thank you so much!
To add to Mr F's great hints, here is something :
How can you know by just one sight that it converges or diverges ? How can you know if you have to find a diverging sequence that is inferior or a converging sequence that is superior ?
Study the asymptotic equivalence. This may sound a rude word, but here it is. Let's take
When n goes very large, 2 can be neglected. More formally, you can write :
So when n goes large,
Now, you should know you have to compare it to the Riemann series :
converges if and only if
Is ? No. Hence it diverges.
When it comes to stuff with , remember that when n is very large, the most powerful stays alive :
No need for Leibniz's Criterion since this series fails the most rudimentary test.(i) - Alternating Series??
therfore it diverges by the n-th term test.
If we write this as(ii)
We realize that this is a telescoping series with sum
I prefer root test, but ratio test is acceptable here.(iii) - I suppose you use Ratio test? But I have troubles simplifying this one!
So the series converges
Maybe I can write a 175 letters text on why it's pointless to use the root test when the ratio test works perfectly, avoiding to use any Stirling approximations or whatever unknown limits.
But it would not be a nice text
That's good you don't like l'Hospital's rule anymore