Math 231 Worksheet #8 Solutions
1. This problem concerns the curve C dened by
y = 3 3x
y
0x1
(a) Sketch the curve and use geometry to compute its length.
Solution 1a:
From Pythagorean theorem: L2 = 32 + 12 gives L = 10
3
B
2 B 3 3x
B
1 B
B
1 2
x
(b) Use i

Math 231 Worksheet #9 Solution
1. Let cfw_an be a sequence.
(a) What does it mean to write an L?
Solution:
an L means limn an = L. The sequence an has the limit L so for large
values of n, the an are very close to L.
(b) If an L does the new sequence cfw

Math 231 Worksheet #11 Solutions
1. A series has a sum if what sequence has a limit?
Solution:
an = a1 + a2 + . . . + an , let sn denote the nth partial sum
Given a series
n=1
s n = n ai = a1 + a2 + . . . + an .
i=1
If the sequence cfw_sn is convergent a

Math 231 Worksheet #10 Solutions
1. Recall that the rule of distribution says that for any real numbers
X(C0 + C1 ) = XC0 + XC1
(A + B)Y = AY + BY
(a) By rst letting X = (A + B), show the distribution rule
(A + B)(C0 + C1 ) = AC0 + AC1 + BC0 + BC1
Solutio

Math 231 Worksheet #13 Solutions
1
1. (a) Use the comparison test with n2 to show n21 converges.
n=1
n=1
+1
Solution:
1
n21 for all n 1
n2
+1
1
We know that
n=1 n2 converges by the integral test, so the original series
1
n=1 n2 +1 must also converge.
1
(

Math 231 Worksheet #12
1. (Review)
1
a) The series n is called the . . . series. It is convergent/divergent.
n=1
Solution:
Harmonic series. It is divergent.
b) The integral 1 x1p dx is convergent for p such that . . . , and divergent for p such
that . . .

Math 231 Worksheet #14 Solutions
1. What is the dierence between a series absolutely converging, conditionally converging
and diverging?
Solution:
A series
an is called absolutely convergent if the series of absolute values
|an |
is convergent.
A series a

1. (10 points each)
Determine if the series is absolutely convergent, conditionally convergent
or divergent. Be sure to show your reasoning. No work, no credit.
(a)
1
X
n=5
p
1
n3 + 10n
1
X ( 1)n
(b)
6n + 3
n=0
(c)
P1
n=2 (
1)n n+2
n
2
2. (10 points each)