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The Taylor Series


Differentiation and Integration of Power Series

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Problem : Prove that (sin(x)) = cos(x) using the Taylor series of the two functions.


sin(x) = x 2n+1    

has infinite radius of convergence, we can differentiate term-by-term:

(sin(x)) = x 2n+1  
  = x 2n  
  = x 2n  
  = cos(x)  

Problem : Prove that 1/(x + 1)dx = log(x + 1) using Taylor series. (The Taylor series for 1/(x + 1) is (- 1)n x n .)

We have

dx = (- 1)n x n dx  
  = x n+1  
  = log(x)