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The Taylor Series

Math

Problems 4

Summary Problems 4

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

Since

sin(x) = x2n+1    

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


(sin(x))=x2n+1  
 =x2n  
 =x2n  
 =cos(x)  

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

We have


dx=(- 1)nxndx  
 =xn+1  
 =log(x)