Understanding Wikipedia Raabe's test proof

Question

I am having a hard time understanding the following proof on Wikipedia. This is the second question I post regarding steps in the proof.

Question:

1) When it is proven for $R<1$ that $a_{n+1}\geqslant \frac{ca_{N}}{n^R}$. The series are said to be divergent because $a_{n+1}\geqslant 0$ since $\frac{ca_{N}}{n^R}$ is bigger than zero.

2) What is being used in the second part of the proof? How can $O(\frac{1}{n^2})$ become $O(1)$?

Thanks in advance!

Answer 1

(1) No, it's not divergent because $a_{n++1}\ge0$, that's nonsense. It's divergent because $\sum1/n^R$ is divergent for $0<R<1$.

(2) The $O(\frac1{n^2})$ did not "become" $O(1)$. We had a sequence of equations, with error term $O(1/n^2)$. We added those equations. The error term in the sum is the sum of the error terms. Since $\sum 1/n^2$ converges, the error term in the sum is $O(1)$.

Answer 2

1) We have series with positive terms, and $\frac{ca_N}{n^R}$ diverges if $R\le 1$, so they use the comparison test.