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its Taylor Series? That is, does
∞ f (x) =
n=0 f (n) (a)
(x − a)n ?
n! Unfortunately, we saw that this need not be true even if the Taylor Series converges everywhere. Taylor’s Theorem gives us a means to show that for many important functions this
equality does hold. To see why this is the case, note that if we ﬁx an x0 , then
k Sk (x0 ) =
n=0 f (n) (a)
(x0 − a)n
n! is not only the k -th partial sum of the Taylor Series for f (x) centered at x = a, but it is
also the k -th degree Taylor polynomial. As such, Taylor’s Theorem shows that
| f (x) − Sk (x0 ) |=| Rk,a (x0 ) |
If we can show that
lim Rk,a (x0 ) = 0 k→∞ then we get ∞ f (x) = lim Sk (x0 ) =
k→∞ n=0 f (n) (a)
(x − a)n .
n! Therefore, f (x) agrees with its Taylor series precisely when the Taylor remainders
Rk,a (x) → 0
DE Math 128 335 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series Before we present the ﬁrst example of this section, we will need to recall the following l...

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