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QUESTION


Show that when $0<|z|<4$ $${1\over
4z-z^2}={1\over4z}+\sum_{n=0}^{\infty}{z^n\over 4^{n+2}}.$$



ANSWER


${1\over 4z-z^2}={1\over z(4-z)}={1\over 4}({1\over z}+{1\over
4-z})$ (partial fractions)$={1\over 4z}+{1\over 4^2}(1-{z\over
4}={1\over 4z}+\sum_{n=0}^\infty {z^n\over 4^{n+2}}$. (This is the
Laurent expansion for $0<|z|<\infty.)$




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