Using Taylor theorem prove that x-((x^3)/6)<sinx<x-((x^3)/6)+((x^5)/120) for x>0

Question

Given that x>0 and to prove it using Taylor series 

Answer 1

The infinite Maclaurin series for sine is well known (and it has alternating signs from term to term).  (Special case of Taylor series for a=0)

 

Referring to your 3-part inequality, the left side is that series truncated to 2 terms; the right side is that series truncated to 3 terms.

 

Since, in the case of the left side, the omitted terms decrease in absolute value, uniformly (abs. values are successively smaller & smaller), and the first of those omitted terms would be positive, the left-side 'estimate' is necessarily < the true sin x value.

 

Since, in the case of the right side, the omitted terms decrease in absolute value, uniformly, and the first of those omitted terms would be negative, the right-side 'estimate' is necessarily > the true sin x value.