Is this sequence decreasing? If a sequence $b_n&gt;0$ and $b_n$ converges to $0$, can we say it is eventually decreasing? This problem bumps up when I am trying to something bigger. However, I am very unsure of this. If this is not true, can anyone give some counter examples please?
Here's one way to construct lots of counterexamples.  Take any positive term sequence, $(a_n)$ that is strictly decreasing to $0$.  Then form the new sequence $a_2,a_1,a_4,a_3,a_6,a_5,\dots$. (I.e., you swap each even-indexed term with the preceding odd-indexed term.)
If a series is convergent then it will satisfy the following:

$$
\lim_{n \to \infty} a_n = 0 \\
\lim_{n \to \infty} \frac {a_{n+1}}{a_n} \le 1
$$

So for some very large $n$ you could say that the series is decreasing, because $a_{n+1} \le a_n$, but since consecutive terms might be equal it depends on what "decreasing" is defined to mean.