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8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
We can apply the main idea from Example 8.3 in general. To determine the values of
x for which a Taylor series
∞
k=0
c k (x − a)
k ,
centered at x = a will converge, we apply the Ratio Test with a k = |c k (x − a) k | and recall
that the series to which the Ratio Test is applied converges if lim k→∞
a k+1
a k
< 1.
Observe that
a k+1
a k
= |x − a|
|c k+1 |
|c k |
,
so when we apply the Ratio Test, we get that
lim
k→∞
a k+1
a k
= lim
k→∞
|x − a|
c k+1
c k
.
Note further that c k =
f (k) (a)
k! , and say that
lim
k→∞
c k+1
c k
= L.
Thus, we have found that
lim
k→∞
a k+1
a k
= |x − a| · L.
There are three important possibilities for L: L can be 0, a finite positive value, or infinite.
Based on this value of L, we can therefore determine for which values of x the original
Taylor series converges.
• If L = 0, then the Taylor series converges on (−∞, ∞).
• If L is infinite, then the Taylor series converges only at x = a.
• If L is finite and nonzero, then the Taylor series converges absolutely for all x that
satisfy
|x − a| · L < 1.
In other words, the series converges absolutely for all x such that
|x − a| <
1
L
,
which is also the interval
a −
1
L
, a +
1
L
.
Because the Ratio Test is inconclusive when the |x − a| · L = 1, the endpoints a ±
1
L
have to be checked separately.
It is important to notice that the set of x values at which a Taylor series converges