438 Formulas and Theorems
(b) Givenr=f(θ)andα≤θ≤β, the area
of the region between the curve, the origin,
θ=αandθ=β:
A=
∫β
α
1
2
r^2 dθ or A=
1
2
∫β
α
[f(θ)]^2 dθ.
(c) Area between two Polar Curves:
Givenr 1 = f(θ)andr 2 =g(θ),
0 ≤r 1 ≤r 2 andα≤θ≤β, the area
betweenr 1 andr 2 :
A=
∫β
α
1
2
(
r 2
) 2
dθ−
∫β
α
1
2
(
r 1
) 2
dθ
=
∫β
α
1
2
((
r 2
) 2
−
(
r 1
) 2 )
dθ.
- Series and Convergence:
(a) Geometric Series:
∑∞
k= 0
ark=a+ar+ar^2 +ar^3 +···
+ark−^1 ··· (a=/0)
if|r|≥1, series diverges;
if|r|<1, series converges and the
sum=
a
1 −r
.
(Partial sum of the firstnterms:
Sn=
a−arn
1 −r
for all geometric series.)
(b) p- Series:
∑∞
k= 1
1
kp
= 1 +
1
2 p
+
1
3 p
+
1
4 p
···
+
1
kp
+···
ifp>1, series converges;
if 0<p≤1, series diverges.
(c) Alternating Series:
∑∞
k= 1
(− 1 )k+^1 ak=a 1 −
a 2 +a 3 −a 4 +···+(− 1 )k+^1 ak+···or
∑∞
k= 1
(− 1 )kak=−a 1 +a 2 −a 3 +a 4 −
···+(− 1 )kak+···, whereak>0 for all
ks.
Series converges if
(1) a 1 ≥a 2 ≥a 3 ··· ≥ak≥ ···and
(2) klim→∞ak=0.
(Note: Both conditions must be satisfied
before the series converges.)
Error Approximation:
IfS=sum of an alternating series,andSn=
partial sum of n terms,∣ then
∣error∣∣=|S−Sn|≤an+ 1.
(d) Harmonic Series:
∑∞
k= 1
1
k
= 1 +
1
2
+
1
3
+
1
4
+···diverges.
Alternating Harmonic Series:
∑∞
k= 1
(− 1 )k+^11
k
= 1 −
1
2
+
1
3
−
1
4
+···+
(− 1 )
k+ 11
k
+···converges.
(
∑∞
k= 1
(− 1 )k^1
k
=− 1 +
1
2
−
1
3
+
1
4
−
···+(− 1 )k
1
k
+···also converges.)
- Convergence Tests for Series:
(a) Divergence Test:
Given a series
∑∞
k= 1
ak, if limk→∞ak=/0, then the
series diverges.
(b) Ratio Test for Absolute Convergence:
Given
∑∞
k= 1
akwhereak/=0 for allks and let
p=klim→∞
|ak+ 1 |
|ak|
, then the series
∑∞
k= 1
ak
(1)converges absolutely ifp<1;
(2)diverges ifp>1;
(3)needs more testing ifp=1.
(c) Comparison Test:
Given
∑∞
k= 1
akand
∑∞
k= 1
bkwith
ak>0,bk>0 for allks, and
a 1 ≤b 1 ,a 2 ≤b 2 ,...ak≤bkfor allks:
(1)If
∑∞
k= 1
bkconverges, then
∑∞
k= 1
ak
converges.
(Note that if the bigger series converges,
then the smaller series converges.)