8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
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of x for which a given Taylor series converges? Moreover, given the Taylor series for a
function f , does it actually converge to f (x) for those values of x for which the Taylor
series converges?
Example 8.3. Graphical evidence suggests that the Taylor series centered at 0 for e x
converges for all values of x. To verify this, use the Ratio Test to determine all values of x
for which the Taylor series
∞
k=0
x k
k!
(8.21)
converges absolutely.
Solution. In previous work, we used the Ratio Test on series of numbers that did not
involve a variable; recall, too, that the Ratio Test only applies to series of nonnegative
terms. In this example, we have to address the presence of the variable x. Because we are
interested in absolute convergence, we apply the Ratio Test to the series
∞
k=0
x k
k!
=
∞
k=0
|x| k
k!
.
Now, observe that
lim
k→∞
a k+1
a k
= lim
k→∞
|x| k+1
(k+1)!
|x| k
k
= lim
k→∞
|x| k+1 k!
|x| k (k + 1)!
= lim
k→∞
|x|
k + 1
= 0
for any value of x. So the Taylor series (8.21) converges absolutely for every value of x,
and thus converges for every value of x.
One key question remains: while the Taylor series for e x converges for all x, what we
have done does not tell us that this Taylor series actually converges to e x for each x. We’ll
return to this question when we consider the error in a Taylor approximation near the end
of this section.