The sequence $ \{a_n \} $ defined recursively as,
$ a_1 = 1, a_2 = 2 $ and $ a_ {n + 2} = \dfrac{1}{2} (a_n + a_ {n + 1}) $ for $ n> 3 $.
I tried and found that the limit of that series is given by $ \dfrac {2} {3} \cdot 2+ \dfrac {1} {3} $ but I don't know if that result is good.
You can use characteristic equation to solve it. $$2x^2-x-1=0 \implies x = 1 \text{ or } x = -\frac{1}{2}$$ Therefore $a_n = (1)^n \cdot c_0+(-\frac{1}{2})^n\cdot c_1 \implies 1 = a_1 = c_0-\frac{1}{2}c_1$ and $2 = a_2=c_0+\frac{1}{4}c_1 \implies c_0=\frac{5}{3}$ and $c_1=\frac{4}{3}$, which means that $$a_n=\frac{5}{3}+(-\frac{1}{2})^n\cdot\frac{4}{3} \implies \lim_{n \to\infty}(a_n)=\frac{5}{3}$$
$$a_{n+2}-a_{n+1} = -\frac 12 (a_{n+1}-a_n) \implies a_{n+1}-a_n=\left(-\frac 12\right)^{n-1} (a_2-a_1) = \left(-\frac 12\right)^{n-1}\tag 1 $$
$$a_{n+2}+\frac 12 a_{n+1} = a_{n+1} + \frac 12 a_n \implies a_{n+1} + \frac 1 2 a_n= a_2+\frac 12 a_1 = \frac 52 \tag 2 $$
(2) - (1) $$ a_n = \frac 23 \left(\frac 52 - \left(-\frac 12 \right)^{n-1} \right) $$
Check here for an explanation: Finding explicit formula for recursive relation
Using a standard method for
solving homogeneous linear recurrence relations with constant coefficients,
it can be seen that $a_n=\dfrac53+\dfrac43\left(-\dfrac12\right)^n$, so you are correct that $\lim\limits_{n\to\infty}a_n=\dfrac53$.