I have a few questions to ask about series and convergence tests. I have been struggling to study everything fully and if someone can give me advices I will be really thankful.This is what I know so far, and please correct me if I am wrong with anything.
Definition 1. For the series $\sum_{n=1}^\infty a_n$ we say that it converges if its sequence of partial sums converges. If it doesn't converge, we say it diverges. Let's say $S_k$ is the partial sum. If $S_k\to \pm \infty$ we say that the series diverges. If $S_k\to$ nothing , we say it diverges as well.
Defintion 2. $\sum_{n=1}^\infty a_n$, if the series converges then the sequence $(a_n)\to 0$ ($n\to \infty$). If $(a_n)\to 0$ then it does not mean that the $\sum_{n=1}^\infty a_n$ converges.
Examples: $\sum_{n=1}^\infty \frac{1}{n}$ diverges, $\sum_{n=1}^\infty \frac{1}{n^\alpha}$ (if $ \alpha$ >1 converges, else diverges) etc.
How do we do tests for the convergence of the following series:
a) $\sum_{n=1}^\infty \arctan\frac{\sqrt n -2}{\sqrt n + 2}$
b) $\sum_{n=1}^\infty \ln (1+\frac{1}{n})$
I have a few questions for the absolute convergence.
Definition 3. For the series $\sum a_n$ we say that it converges absolutely if the series $\sum |a_n|$ converges. If the series converges but doesn't converge absolutely, we say it conditionally converges.
Definition 4. LEIBNIZ: (alternate series) $\sum (-1)^n \cdot b_n$ , if $b_n \to 0$ , the series converges.
ABELL: $\sum a_n b_n $, if 1) $\sum a_n$ converges
2) sequence $(b_n)$ is decreasing/increasing and bounded, then $\sum a_n b_n $ converges.
How do we check the absolute and normal convergence of the following series:
a) $\sum_{n=1}^\infty \frac{1}{\sqrt[3]{n^2+1}}\cdot \sin \frac{n\pi}{3}$
b) $\sum_{n=1}^\infty (-1)^n \cdot \tan \frac{3}{\sqrt[4]{n}}$
Thanks.