find power series expansions for the following functions: (i) f(x)=1/(x+1). Let x = -t; then 1...

 

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find power series expansions for the following functions: (i) f(x)=1/(x+1). Let x = -t; then 1 1 - I 1 ----- 1+2 = (-2)=(-1)+2 for x=-t2 <1. Therefore, 1 x+1 = n=0 n=0 (-1), for all |x| < 1. n=0 (ii) f(x)=1/x. Let x=-(t-1); then 1 = 1-x t n=0 8 - -(-1))=(-1)" (t-1)", n=0 for x=-(t-1)| < 1. Therefore, (iii) f(x) = 1/(1-x). 1 - = I n=0 (-1)(x-1)", for all |x 1| < 1. Since 1/(1-x) is the derivative of 1/(1-x), from Theorem 4.15, find power series expansions for the following functions: (i) f(x)=1/(x+1). Let x = -t; then 1 1 - I 1 ----- 1+2 = (-2)=(-1)+2 for x=-t2 <1. Therefore, 1 x+1 = n=0 n=0 (-1), for all |x| < 1. n=0 (ii) f(x)=1/x. Let x=-(t-1); then 1 = 1-x t n=0 8 - -(-1))=(-1)" (t-1)", n=0 for x=-(t-1)| < 1. Therefore, (iii) f(x) = 1/(1-x). 1 - = I n=0 (-1)(x-1)", for all |x 1| < 1. Since 1/(1-x) is the derivative of 1/(1-x), from Theorem 4.15,