(a) Find the value of the convergent series
∑ n = 2 ∞ ( 2 n + 1 3 n + 1 2 n − 1 − 1 2 n + 1 ) {\displaystyle \sum _{n=2}^{\infty }\left({\frac {2^{n+1}}{3^{n}}}+{\frac {1}{2n-1}}-{\frac {1}{2n+1}}\right)}
(simplify your answer completely).
