MATH 141H Exam 4 Preparation
Office Hours
(1) Tuesday & Thursday : 9am – 4pm by appointment only
(2) Wednesday: 2 – 4pm,
Office: 4123 CSIC,
(3) Friday: 9 – 10:30 am
Email: [email protected]
1. Infinite sequence
(1) Find f (x) such that f (n) = an . Then, lim an = lim f (x)
n→∞
(i) L’Hospital’s rule.
x→∞
(ii) Simplify an if necessary.
(2) The comparison property
(3) Bounded sequence, Increasing/decreasing sequence
(i) If the sequence an is bounded and either increasing and decreasing, then lim an exists.
n→∞
(ii) A sequence defined recursively: proof by induction (For examples, see Homework 11)
2. Infinite series
(1) Convergence of series by using partial sums Sn =
n
X
ak
k=1
(i) Telescope series: cancellation.
∞
X
(ii) Geometric series:
n=m
(2) Divergence test: If lim an 6= 0, then
n→∞
∞
X
crn =
crm
for |r| < 1
1−r
an diverges.
n=1
(3) Convergence test for positive series:
∞
X
an , an > 0
n=1
(i) The ratio test, the root test, the integral test, the comparison test
(4) If an > 0, decreasing and lim an = 0, then the series
n→∞
(5) Absolute convergence test:
∞
X
∞
X
(−1)n an converges.
n=1
|an | converges, then
n=1
(i) Absolute convergence/ conditional convergence
∞
X
(−1)n an converges.
n=1
3. Power series
∞
X
cn x n
n=1
00
f (0) 2
f (n) (0) n
(1) Taylor polynomial: pn (x) = f (0) + f (0)x +
x + ···
x
2!
n!
0
(2) Radius of convergence by generalized convergence test
(3) The interval of convergence: radius of convergence & the convergence at the end points
(4) Known cases
(i) ex =
∞
X
xn
n=0
(ii)
∞
X
n!
xn =
n=m
for all x ∈ R
xm
for |x| < 1
1−x
(5) Differentiating and integrating power series.
4. Complex numbers
√
a2 + b2 . Quotients of complex numbers.
√
(2) Polar decomposition: z = a + ib =⇒ z = Reiθ , R = a2 + b2 . The argument θ is determined
from the picture in the plane.
(1) z = a + ib =⇒ z̄ = a − ib, |z| =
(3) nth roots of a complex number, and picture of such numbers in the plane.
(4) Complex power series
z
(i) e =
∞
X
zn
n=0
(ii)
∞
X
n=m
n!
zn =
for all z ∈ C
zm
for |z| < 1
1−z
5. Review homework problems
1. Evaluate the limit.
(1)
(2)
(3)
(4)
(5)
lim
n→∞
p
p
2n2 + 7n − 2n2 + n
lim e2n − 1
1
n
n→∞
lim
n→∞
1−
e n
n
1
lim (3n) n
n→∞
lim
1+
1
2
+
n→∞
1
3
+ ··· +
ln n
1
n
2. Determine whether the series converges or diverges. Name the test you use.
(1)
∞
X
(−1)n
n=1
(2)
(3)
n
4n−1
∞
X
cos(nπ)
√
n
n=1
∞
X
√
4 n
(−1)n 5
n
n=1
(4)
∞
X
(−1)n
n=1
ln n
n
3. Compute the series.
(1)
∞ n+3
X
3
n=1
(2)
∞
X
n=2
5n−1
+ (−1)n
4
(n + 1)(n + 5)
22n
n
(25) 2
4. Find the radius of convergence of the series.
(1)
∞
X
2n
n=1
(2)
∞
X
n!
xn
n5 3n x2n
n=1
(3)
√
∞
X
nn n
n=1
(4)
∞
X
n!
zn
(1 − 2i)n z n
n=1
5. Find the interval of convergence of the series.
(1)
∞
X
(−1)n
n=1
(2)
x2n+1
n
∞
X
(x + 1)2
n(n + 1)
n=1
(3)
∞
X
n+1
n=1
n2
xn
6. Find the power series for the function.
(1)
f (x) =
1
(x + 1)2
(2)
f (x) =
1
4 + x2
(3)
f (x) =
x3
x2 + 2x + 2
Z
(4)
x
t ln(1 + t)dt
0
7. Find the 3rd Taylor polynomial p3 (x) about x = 0 of f (x) = tan(3x).
8. Write the complex number/series in a + ib form or Reiθ form.
(1)
(2)
2 + 4i
in a + ib form.
1+i
∞ X
1
i n
in a + ib form and Reiθ form.
+
2 2
n=0
(3)
(4)
√
(−2 + 2 3i)31 in a + ib form.
√
(1 + 3i)5 in Reiθ form.
9. Find the nth roots of the complex number and draw all of those points in the plane.
(1)
w = 16,
n=8
(2)
w = −1 + i,
n=3