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**Unformatted text preview: **Then
∞ f (x) =
n=0 f (n) (a)
(x − a)n
n! for all x ∈ I .
Example.
Let f (x) = ex and let a = 0. Let I = [−B, B ]. We know that for eack k , f (k) (x) = ex .
Moreover, since ex is increasing,
0 < e−B ≤ ex ≤ eB
for all x ∈ [−B, B ]. This means that if M = eB , then for all x ∈ [−B, B ] and all k , we
have
| f (k) (x) |= ex ≤ eB = M.
We have satisﬁed all of the conditions of the Convergence Theorem for Taylor Series. It
follows that for any x ∈ [−B, B ],
∞
x e=
n=0 f (n) (0) n
(x) =
n! ∞ n=0 xn
n! Finally, we see that this would work no matter what B we choose. However, given any
x ∈ R, if we pick a B such that | x |< B , then x ∈ [−B, B ]. This means that for this x
∞
x e=
n=0 DE Math 128 337 xn
.
n!
(B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series Hence for every x ∈ R, the equality
∞ ex =
n=0 xn
n! holds. 4.5.4 Some Additional Applications
of Taylor Series In this section we will present some further examples of functions that are representable
by their Taylor series and see what this tells us about these functions.
Example.
Find a power series representation for...

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