Write-ups due - extended to Tuesday Oct. 3:
discussed in lecture today: Sec.
2.3.2 in Bounds
of sets of numbers
and Appendix B.2 (p.849, 850, top of 851)
(see again the top of the page with exercises).
write up and turn in: Exercises above (p.6) 2.3.2, 2.3.3
Prove by induction formulas (4) and (5) page 851
discussed in lecture today: induction,
(web resource -
Useful formula: the sum of the
first n terms of an arithmetic progression with common difference d is
+...+ an = n a1
+ d n(n-1)/2.
Show, using induction, that the general term of a geometric
of common ratio r is
also 1. Which
of the sums in problem 1a,b,c [are
sums of] arithmetic progressions? Find the first term and the
= b1 r^(n-1) Notation: b^n
denotes b to the power n
write up and turn in: B.1 problems 1a,b,c, 4b, 8
and calculate the sums using the formula for the sum of the first n
terms of an arithmetic progression.
2. Calculate the sums (note that the
terms are geometric progression!):
a. 1+3+3^2 +3^3+...+3^n
topics: calculation of (some type of) sums.
write up and turn in: from the textbook, Sec. B1 (page 854) 1c,d,e (you may use any method)
1/2 -1/4 +1/8 - 1/16 +...+ (-1)^(n+1)/2^n . Is the general
term correctly written?
(2n+1) + (2n+2) + (2n+3) +...+ (4n).
Thursday Sept 28
sums, review the absolute value, start Sec. 2.5
Reminder: the distance between
the numbers a and b plotted on the number line is |a-b|.
Solve: if you need more
practice with sums, try the following:
calculate the sum in problem B1 4a; calculate
calculate 1+2+3+...+(8n+3); calculate 101+102+103+...+n
(where n>100, for the sum to make sense).
write up and turn in: Consider the function f(x)=(4x^2-4)/(x-1).
1. What is the domain of this function?
2. Plot the function. (Note that for x not equal to 1 the
fraction can be simplified, yielding 4x+4.)
By inspecting the graph, it appears that f(x) is as close to 8
as we wish provided that x is close enough to 1 (but not 1).
Let us check this:
3. Show that |f(x)-8| < .1 (so: the distance between
f(x) and 8 is smaller than .1) provided that
< .1 /4 and x not 1(so: the distance between x and 1 is smaller than
.1/4, but not 0).
4. |f(x)-8|< .00001 provided that x is how close to 1(but
5. Let us denote by epsilon a very, very small positive number
(think like .00000000001, or like 10^(-10,000), or maybe even
Then |f(x)-8|<epsilon provided that x is how close to 1 (but not 1)?
Bonus problem (extra 5 points): Prove
the triangle inequality: |a+b|<|a|+|b|
tutor locations and
Topcs: 2.5, A.2 Theorems
1, 2(proved), 3(proved), 4(proved i), 5.
write up and turn in: 1. Use the epsilon-delta definition of
limits to show that if a function f(x) has the limit L as x->a
(write for short ' f(x)->L as x->a ') and if c is a
constant, then cf(x)->cL as x->a. (This is
the last property we discussed in class today.)
2. Use the properties listed
in class (see A.2 Theorems 1,2,3,4,5) to show that the following are
true as as x->a :
a) x^2 -> a^2
b) x^3-> a^3
c) x^n-> a^n (use induction!)
3. Use the properties listed in class and problem 2. to argue* that if P(x) and Q(x)
are polynomials, then
a) P(x) ->
P(a) as x->a
b) P(x)/Q(x) ->
x->a if Q(a) is not zero.
*In this problem 'argue' means that you can use '... ' and
you are spared a proof by induction (unlike 2.c, where the property is clear
using '...', yet you are asked to
use induction nevertheless).
Bonus problem: (extra 3 points)
Use induction to show that |a1+a2+...+an| <| a1|+|a2|...+|an|.
Write-ups of the problems above: due date extended to
Tuesday Oct. 3