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∑ k = 0 ∞ ( − 1 ) k α + β k = ∑ k = 0 ∞ ( 1 α + β ⋅ 2 k − 1 α + β ( 2 k + 1 ) ) {\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{\alpha +\beta \,k}}=\sum _{k=0}^{\infty }\left({\frac {1}{\alpha +\beta \cdot 2k}}-{\frac {1}{\alpha +\beta \,(2k+1)}}\right)} = 1 2 β ∑ k = 0 ∞ ( 1 α 2 β + k − 1 1 2 + α 2 β + k ) = 1 2 β [ ψ ( 1 2 + α 2 β ) − ψ ( α 2 β ) ] {\displaystyle ={\frac {1}{2\beta }}\sum _{k=0}^{\infty }\left({\frac {1}{{\frac {\alpha }{2\beta }}+k}}-{\frac {1}{{\frac {1}{2}}+{\frac {\alpha }{2\beta }}+k}}\right)={\frac {1}{2\beta }}\left[\psi \left({\frac {1}{2}}+{\frac {\alpha }{2\beta }}\right)-\psi \left({\frac {\alpha }{2\beta }}\right)\right]}
![{\displaystyle ={\frac {1}{2\beta }}\sum _{k=0}^{\infty }\left({\frac {1}{{\frac {\alpha }{2\beta }}+k}}-{\frac {1}{{\frac {1}{2}}+{\frac {\alpha }{2\beta }}+k}}\right)={\frac {1}{2\beta }}\left[\psi \left({\frac {1}{2}}+{\frac {\alpha }{2\beta }}\right)-\psi \left({\frac {\alpha }{2\beta }}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65edac5b38011bdd25c8bd690b0f5b2b03b286cc)
![{\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{\alpha +\beta \,k}}={\frac {1}{2\beta }}\,\left[\psi \left({\frac {1}{2}}+{\frac {\alpha }{2\beta }}\right)-\psi \left({\frac {\alpha }{2\beta }}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41fe42f2d96af3eccde615b236b6b85be896aac8)
