By employing the coefficient extraction method from hypergeometric series, we shall establish numerous closed form evaluations for infinite series containing central binomial coefficients and harmonic numbers, including several conjectured ones made by Z.-W. Sun.

By applying the "coefficient extraction method" to the symmetric transformation of hypergeometric series due to Chu and Zhang (2014), we examine systematically five classes of infinite series of convergence rate " $-1/4$" containing binomial coefficients and harmonic numbers. Numerous closed formulae in terms of universal constants (such as $\pi$, $\ln 2$, and the Riemann zeta values) are established.

Three classes of infinite series containing binomial coefficient $(\genfrac{}{}{0ex}{}{3n}{n})$, harmonic-like numbers, and an independent variable " $y$" are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert's series and manipulating the cubic transformations for the ${}_3{F}_2$-series through the "coefficient extraction" method.

By making use of the multisection series method, four classes of alternating infinite series are evaluated, in closed form, by the Riemann zeta function and the Dirichlet beta function.