5 Steps to a 5 AP Calculus BC 2019

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.

35. 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.