By Exercise 86, Ta (x) converges for |x| < 1, but we do not yet know...

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By Exercise 86, Ta (x) converges for |x| < 1, but we do not yet know whether Ta(x) = (1+x)ª. (a) Verify the identity a - () -- () + - + (₁ : 1) D n+ (b) Use (a) to show that y = Ta(x) satisfies the differential equation (1+x)y' = ay with initial condition y(0) = 1. (c) Prove that Ta(x) = (1+x) for x < 1 by showing that the derivative of the ratio Ta(x) is zero. (1+x)a By Exercise 86, Ta (x) converges for |x| < 1, but we do not yet know whether Ta(x) = (1+x)ª. (a) Verify the identity a - () -- () + - + (₁ : 1) D n+ (b) Use (a) to show that y = Ta(x) satisfies the differential equation (1+x)y' = ay with initial condition y(0) = 1. (c) Prove that Ta(x) = (1+x) for x < 1 by showing that the derivative of the ratio Ta(x) is zero. (1+x)a

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