Taylor's Theorem: Lagrange Form for Approximating Error

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The discussion centers on the utility of the Lagrange form of Taylor's Theorem for approximating error despite the integral form providing exact error values. Participants note that while common remainder forms like Cauchy, Lagrange, and Schlömilch are not precise, they serve practical purposes. The complexity of calculating exact answers makes simple approximations more beneficial in many cases. The consensus highlights the value of ease of use over exactness in certain mathematical applications. Overall, simpler approximations can often yield more practical results than complicated exact calculations.
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If the integral form of the remainder term gives the exact error, why do we use the Lagrange form of the remainder to approximate the error.
 
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All the common forms of the remainder Cauchy, Lagrange and Schlömilch; are not helpful in exact form. The exact answer is hard to compute. A simple approximation often is more useful than a complicated exact value.
 
Oh! that's so simple, thanks.
 

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