Inferences about convergence based on two pieces of data

Question

I am new to calculus and just wanted to check my reasoning:

$\sum a_n$ is positive and converges.

This means that:

1. $\sum (-1)^na_n$ also converges, because it converges absolutely.
2. $\sum(sin(n)-2cos(n))·a_n$ converges, because $sin(n)-2cos(n)$ fluctuates between +2.23607 and -2.23607, therefore functioning much like a constant that could be moved to the right of the sum sign.
3. $\sum a_n x^n$ converges if and only if $-1\leq x \leq 1$
4. $\sum a_n^2$ converges, because the data given implies that $lim_{n\to\infty}a_n=0$: therefore $a_n<1$ and $a^2_n<a_n$
5. $a_n$ is a decreasing series (at least starting from some point), again because the data given implies that $lim_{n\to\infty}a_n=0$.

Are each of these conclusions correct?

Thank you!

Answer 1

1) First one is true.

2) Second one also, even though the more convenient way to do the majoration is to write $$|(\sin(n)-2\cos(n))a_n| \leq 3a_n$$

3) This one is not true. For example, let's take $a_n = \frac{1}{3^n}$. Then $\sum a_n x^n$ converges for $x=2$.

4) This is true.

5) This, I don't understand. The sequence $(a_n)$ converges to $0$, but you can say nothing about its monotony. And the series $\sum a_n$ is increasing, because all the $a_n$ are positive.

Answer 2

Number 5 is wrong, consider $a_n=\begin{cases}\frac{1}{n^2} \text{ if $n$ is even}\\ \frac{1}{n^3} \text{ if $n$ is odd}\end{cases}$