Question

Ratio test question: if I have a series and I have to apply ratio test to consider whether it converges or not, does it matter if I switch the order of the test? I meant \(\lim |\dfrac{a_{n}}{a_{n+1}}|\) instead of \(\lim|\dfrac{a_{n+1}}{a_n}|\)??
If I can't, in what case I can?

Answer 1

Say you get the normal way is 1/2<1 which says the series a_n converges but the other way we would get the limit is 2>1...so if you you flip things when it comes to the limit thing then you would also have to change your inequalities about that result also what if normal way gives you 0 then reciprocal way gives you inf or - inf. Or what if we get vice versa that. Just a few things to consider.

Answer 2

Thanks for making it clear. How about Weierstrass M test. Can you explain me please?

Answer 3

2/1 + 4/1 +8/1 ... this series diverges but | an / a_n+1 | = 1/2

Answer 4

I have to prove \(f(x) =\sum_{n=1}^\infty \dfrac{x^n}{n^2}\) continuous on [0,1]

Answer 5

To do that, just use Weiertrass M test to prove the series converges but I am not quite sure how to

Answer 6

oh, I got it. :) thanks though

Answer 7

sorry im not familiar with that test.

Answer 8

Let \(M_n =\dfrac{a^n}{n^2}\) hence \(M_{n+1}= \dfrac{a^{n+1}}{(n+1)^2}\)
By ratio test, we have lim = a and on [0,1], a< 1, hence \(M_n\) converges that is f(x) converges and then continuous