440 Formulas and Theorems
Partial Sum:
Pn(x)=
∑n
k= 0
f(k)(a)
k!
(x−a)k
=f(a)+f′(a)(x−a)
+
f′′(a)
2!
(x−a)^2 +···
+
f(n)(a)
n!
(x−a)n
Rn(error for Pn(x))=
f(n+^1 )(c)
(n+ 1 )!
(x−a)n+^1 ,if
x>a,c∈(a,x),orifx<a,c∈(x,a),
or ifx=a,c=a.
- Testing a Power Series for Convergence
Given:
∑∞
k= 0
ck(x−a)k=c 0 +c 1 (x−a)+c 2 (x−a)^2
+···+ck(x−a)k+···
(1) Use Ratio Test to find values ofxfor
absolute convergence.
(2) Exactly one of the following cases will
occur:
(a)Series converges only atx=a.
(b)Series converges absolutely for all
x∈R.
(c)Series converges on all
x∈(a−R,a+R)and diverges for
x<a−Rorx>a+R.At the
endpointsx=a−Randx=a+R, use
an Integral Test, an Alternating Series
Test, or a Comparison Test to test for
convergence.