Taylor Series

# Example the function f x ex is particularly well

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Unformatted text preview: val [−π, π ]. Note again the scale for the y -axis. It is clear that near 0, T13,0 (x) and sin(x) are essentially indistinguishible. In fact, we will soon be able to show that for x ∈ [−1, 1], | sin(x) − T13,0 |< 10−12 while for x ∈ [−0.01, 0.01], | sin(x) − T13,0 |< 10−42 . Indeed, in using T13,0 (x) to estimate sin(x) for very small values of x, round-oﬀ errors and the limitations of the accuracy in ﬂoating-point arithmetic become much more signiﬁcant than the true diﬀerence between the functions. Example. The function f (x) = ex is particularly well suited to the process of creating estimates using Taylor polynomials. This is because for any k , the k -th derivative of ex is again ex . This means that for any n, n Tn,0 (x) = k=0 n = k=0 n = k=0 DE Math 128 f k (a) (x − a)k k! e0 (x − 0)k k! xk . k! 323 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series In particular, T0,0 (x) = 1, T1,0 (x) = 1 + x, T2,0 (x) = 1+x+ T3,0 (x) = T4,0 (x) = x2 , 2 x2 1+x+ + 2 x2 1+x+ + 2 x...
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