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Two summation theorems concerning the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mmultiscripts><mi>F</mi><mn>1</mn><mn>2</mn></mmultiscripts><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate “<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>” may serve as a reference source for readers.