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By employing the bisection series approach, two classes of nonterminating ₃<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mn>2</mn></msub><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-series are examined. Several new summation formulae are established in closed form through the summation formulae of Gauss and Kummer for the ₂<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>±</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-series. They are expressed in terms of well-known functions such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>, Euler–Gamma, and logarithmic functions, which can be used in physics and applied sciences for numerical and theoretical analysis.