Spring 2016 Math 152
Alternating series, ratio test and remainders
5. The Alternating Series Test: The alternating series
∞
P
(−1)n an , where an > 0, converges if it satisfies both
Overview of Material for Test III
courtesy: Amy Austin
n=1
conditions given below:
Integral Teat and Comparison Test
1. The Integral Test:If f (x) is a positive, continuous, de∞
P
ak
creasing function on [1, ∞], and ak = f (k). Then
k=1
Z ∞
and
f (x) dx either both converge or both diverge.
1
• an+1 ≤ an for all n (i.e. the sequence {an } is decreasing).
• lim an = 0
n→∞
6. Def: A series is absolutely convergent if
∞
P
|an | con-
n=1
∞
P
2. The Comparison Test: Suppose
an and
n=1
series of positive terms.
• If
∞
P
∞
P
bn are
bn is convergent and an ≤ bn for all n, then
an
n=1
bn is divergent and an ≥ bn for all n, then
n=1
an converges, but
∞
P
|an | diverges, then
n=1
the series is conditionally convergent.
∞
P
is also convergent. In other words, ”if the larger series
converges, so does the smaller series”.
∞
P
∞
P
n=1
i=1
n=1
• If
verges. If
∞
P
an
n=1
is also divergent. In other words, ”if the smaller series
diverges, so does the larger series”.
7. The Ratio Test:
∞
an+1 = L < 1, then the series P an is abso
• If lim n→∞
an n=1
lutely convergent (and therefore convergent).
an+1 = L > 1 or lim an+1 = ∞, then the
• If lim n→∞
n→∞
an
an ∞
P
series
an is divergent.
n=1
Note: if the larger series diverges, no conclusion can be
made about the smaller series. Likewise, if the smaller
series converges no conclusion can be made about the
larger series.
3. The Limit Comparison Test: If the Comparison Test
is inconclusive, then we may apply the Limit Comparison
Test:
Suppose
∞
P
an and
n=1
∞
P
bn are series of positive terms.
n=1
an
= c > 0, then either both series converge or
bn
both diverge.
n→∞
4. Remainder Estimate: Suppose sn =
n
P
ai = a1 +
i=1
a2 + ...+ an is the nth partial sum of the convergent series
∞
P
an . Then the remainder in using sn to approximate
n=1
the sum S is defined to be Rn = S − sn =
∞
P
ai =
i=n+1
an+1 + an+2 + ....
∞
P
Moreover, if
an was shown to be convergent by the
n=1
integral test where an = f (n), then
Rn =
i=n+1
8. Remainder Estimate and The Alternating Series
Theorem
∞
P
If
(−1)n an , an > 0, is a convergent alternating series,
n=1
If lim
∞
X
an+1 = 1, then the test fails (i.e. is incon
• If lim n→∞
an clusive). Note: If the series does not contain either or a
factorial or an exponential (or both), the ratio test will
be inconclusive.
ai <
Z
and a partial sum
n
P
(−1)i ai is used to approximate the sum of the
sn =
i=1
series with remainder Rn , then |Rn | ≤ an+1 .
Power Series
9. Def: A Power Series is a series of the form
∞
P
cn (x − a)n , where x is the variable and the cn ’s
n=1
are called the coefficients of the series. More generally,
∞
P
cn (x − a)n is called a power series centered at x = a,
n=1
or a power series about a. Specifically,
∞
P
cn xn is a power
n=1
series centered at zero.
∞
n
f (x) dx.
Moreover, the set of all values of x for which the series
converges is called the interval of convergence, de-
1
noted by I. The radius of convergence is R =
of
2
the length of I.
10. Theorem: For a given power series
∞
P
cn (x − a)n there
n=1
are only three possibilities:
(i) The series converges only for x = a, in which case
I = {a} and R = 0.
(ii) The series converges for all x, in which case
15. Known Maclaurin series. Have these Maclaurin series
memorized and know when to use them.
∞
X
1
=
xn = 1 + x + x2 + x3 + x4 ..., provided
(a)
1−x
n=0
−1 < x < 1
∞ xn
P
x2
x3
x4
= 1+x+
+
+
+ ..., provided
2!
3!
4!
n=0 n!
−∞ < x < ∞
(b) ex =
I = (−∞, ∞) and R = ∞.
(iii) There is a positive number R such that the series
converges if |x − a| < R and diverges if |x − a| > R, in
which case I = (a − R, a + R) (test the end points for
convergence) and the radius of convergence is R. In order
to find the radius of convergence, apply the ratio test.
∞ (−1)n x2n
P
x2
x4
x6
= 1−
+
−
+ ...,
(2n)!
2!
4!
6!
n=0
provided −∞ < x < ∞
(c) cos x =
∞ (−1)n x2n+1
P
x3
x5
x7
= x−
+
−
+ ...,
3!
5!
7!
n=0 (2n + 1)!
provided −∞ < x < ∞
(d) sin x =
Representing Functions as Power Series
11. If |x| < 1, then
∞
P
xn =
n=0
12. Theorem: If f (x) =
1
1−x
∞
P
cn xn has a radius of conver-
∞ (−1)n x2n+1
P
x3 x5 x7
= x−
+
−
+ ...,
2n + 1
3
5
7
n=0
provided −1 ≤ x ≤ 1
(e) arctan x =
n=0
gence R, then
a.) f ′ (x) =
∞
P
cn nxn−1 and has a radius of convergence
Taylor Polynomials
n=1
R.
b.)
Z
f (x) dx = C +
convergence R.
∞
X
cn n+1
x
and has a radius of
n+1
n=0
16. Let f (x) be a function. The nth degree Taylor Polynomial for f (x) at x = a is
Tn (x) =
n f (i) (a)
P
(x − a)i
i!
i=0
= f (a)+f ′ (a)(x−a)+
Taylor and Maclaurin Series
13. The Taylor Series of f (x) at x = a is
f (x) =
∞ f (n) (a)
P
(x − a)n
n!
n=0
= f (a)+f ′(a)(x−a)+
f ′′ (a)
f ′′′ (a)
(x−a)2 +
(x−a)3 +...
2!
3!
14. The Maclaurin series is the Taylor series about 0, that
is
∞ f (n) (0)
P
f (x) =
xn
n!
n=0
= f (0) + f ′ (0)x +
f ′′ (0) 2 f ′′′ (0) 3
x +
x + ...
2!
3!
f ′′ (a)
f n (a)
(x−a)2 +...+
(x−a)n
2!
n!
Three dimensional Coordinate System
17. The distance between the points P (x1 , y1 , z1 ) and
Q(x2 , y2 , z2 ) is
p
|P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
18. The midpoint
of the
ing
the
points
P
(x
, y1, z1 )
1
x1 + x2 y1 + y2 z1 + z2
,
,
2
2
2
line
and
segment joinQ(x2 , y2 , z2 ) is
19. The equation of the sphere with center (h, k, l) and
radius r is (x − h)2 + (y − k)2 + (z − l)2 = r2