Given a power series formula, how can I conclude that the coefficients are monotonically decreasing? It would be equally good to conclude that the coefficients are all nonnegative.
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I'm not sure if this is what you're looking for, but the coefficients are completely determined by the successive derivatives at the point about which you're expanding your power series. – Reveillark May 17 '20 at 02:09
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which function are you really looking at – reuns May 17 '20 at 02:48
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The function I am looking at is the reciprocal of exp(bz)-z, where b is a positive parameter and z is the power series variable. The expansion point is zero. – Alan Washburn May 17 '20 at 03:35
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The Taylor coefficients at $0$ of $e^{bz}-z$ are not be decreasing. They will be eventually, but depending on $b>0$ they will increase for a bit. Also, depending on $b$, the coefficient of $z$ can be negative. – May 17 '20 at 07:38
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If b=0, all of the coefficient are 1. If b=1, the first coefficient 1-b is 0. Somewhere between 0 and 1 there is a smallest b where all of the coefficients are nonnegative. What is it? – Alan Washburn May 18 '20 at 18:59