Math 231 Worksheet #8 Solutions
1. This problem concerns the curve C dened by
y = 3 3x
(a) Sketch the curve and use geometry to compute its length.
From Pythagorean theorem: L2 = 32 + 12 gives L = 10
2 B 3 3x
(b) Use i
Math 231 Worksheet #9 Solution
1. Let cfw_an be a sequence.
(a) What does it mean to write an L?
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?
an = a1 + a2 + . . . + an , let sn denote the nth partial sum
Given a series
s n = n ai = a1 + a2 + . . . + an .
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
Math 231 Worksheet #13 Solutions
1. (a) Use the comparison test with n2 to show n21 converges.
n21 for all n 1
We know that
n=1 n2 converges by the integral test, so the original series
n=1 n2 +1 must also converge.
Math 231 Worksheet #12
a) The series n is called the . . . series. It is convergent/divergent.
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
an is called absolutely convergent if the series of absolute values
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.
n3 + 10n
X ( 1)n
6n + 3
2. (10 points each)