# Help with convergence tests for series

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) $$\sum_{n=1}^\infty \frac{1}{\sqrt{n^2+1}}\cdot \sin \frac{n\pi}{3}$$
b) $$\sum_{n=1}^\infty (-1)^n \cdot \tan \frac{3}{\sqrt{n}}$$

Thanks.

• You don't have Dirichlet's convergence test in your list? Feb 16, 2019 at 15:38
• Do you know the limit comparison test( or generally comparison test) ? You should, it is a powerful device Feb 16, 2019 at 20:15
• @MilanStojanovic Trebao bih da znam, s tim se borim najvise. Feb 16, 2019 at 21:09
• @Exzone yadi.sk/i/_3RmSqhr36Ubrn, ovo su zadaci sa resenjima sa MATF u Beogradu, nadam se da ti ne smeta sto je na cirilici Feb 16, 2019 at 21:53
• @MilanStojanovic Hvala ti puno! Naravno da ne smeta :) Feb 16, 2019 at 21:54

Hint for a: I believe that $$lim_{n \rightarrow \infty}arctan(.)=arctan(lim_{n \rightarrow \infty}( \frac{\sqrt n -2}{\sqrt n + 2}))=\frac{\pi}{4}.$$ Therefore, you can deduce a diverges.
Hint for b: The product of positive real numbers $$\prod_{n=1}^{\infty}a_n$$ converges iff the sum $$\sum_{n=1}^{\infty} log(a_n)$$ converges.
If you now look at the sequence of partial products for $$\prod_{n=1}^{M}\frac{n+1}{n}=(M+1)$$, which tends to infinity as M goes to infinity.