Lecture 32 &33 - MATH 118, LECTURES 32 & 33:...

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APPLICATIONS OF TAYLOR SERIES 1 Applications of Taylor Series We have seen how Taylor series can provide a useful approximation of a function on an interval. Furthermore, we have been able to use Taylor’s remainder theorem to bound the maximum error between a function and its Taylor polynomial on an interval. In fact, there are many further applications of Taylor series—it is one of the most useful tools in applied mathematics. The reason is simple: performing many standard mathematical operations (e.g. di±erentiation, integration, taking limits, etc.) is di²cult for many standard functions; however, it is very easy to perform these operations to monomial terms. Since a Taylor Series consists only of terms of this sort, we can often obtain a very good approximation to the desired solution even when an exact solution to a problem eludes us. Sometimes we can even recover the exact solution itself! 1.1 Limits We can often use Taylor series to simplify improper limits, limits of the form “0 / 0” or “ / ”. Previously, we have used l’Hopital’s rule—now we will make use the fact that in the limits x 0 and x → ∞ the monomial terms in the Taylor series have well de³ned behaviours. Example 1: Evaluate the limit lim x 0 sin( x ) x . This is a limit of the form “0 / 0”. Previously, we had evaluated this limit using l’Hopital’s rule; however, since we now know the Taylor series expansion of sin( x ) we can use this to simplify the limit. 1
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lim x 0 sin( x ) x = lim x 0 x - x 3 3! + x 5 5! - ··· x = lim x 0 b 1 - x 2 3! + x 4 5! -··· B = 1 . We recall that by l’Hopital’s rule we had obtained lim x 0 sin( x ) x = lim x 0 cos( x ) 1 = 1 . Example 2: Evaluate the limit lim x 0 x e x - 1 . This is a limit of the form ”0 / 0”. We use the Taylor series expansion of e x to obtain lim x 0 x e x - 1 = lim x 0 x p 1 + x + x 2 2!
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Lecture 32 &33 - MATH 118, LECTURES 32 & 33:...

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