**Write-ups due - extended to Tuesday Oct. 3:**

Monday
Sept 25

and Appendix B.2 (p.849, 850, top of 851)

(see again the top of the page with exercises).

Solve, write up and turn in: Exercises above (p.6) 2.3.2, 2.3.3 d,e,f

Prove by induction formulas (4) and (5) page 851

Tuesday
Sept 26

Topics
discussed in lecture today: induction,

arithmetic and
geometric progressions

(web resource -
see Ch.11)

Useful formula: the sum of the
first n terms of an arithmetic progression with common difference d is

a1 +a2
+...+ an = n a1
+ d n(n-1)/2.

Solve:
Show, using induction, that the general term of a geometric
progression b1,b2,b3,...
of common ratio r is

bn
= b1 r^(n-1) Notation: b^n
denotes b to the power n

Solve,
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

b. 1+r+r^2+...+r^n

c. x+x^2+x^3+...+x^n

d. 1-s+s^2-s^3+...+(-1)^n s^(n)

Wednesday Sept 27

Today's topics: calculation of (some type of) sums.

Solve, write up and turn in: from the textbook, Sec. B1 (page 854) 1c,d,e (you may use any method)

also: 1. Calculate 3+6+9+...+(3n).

2. Calculate 1/2 -1/4 +1/8 - 1/16 +...+ (-1)^(n+1)/2^n . Is the general term correctly written?

3. Calculate (2n+1) + (2n+2) + (2n+3) +...+ (4n).

Thursday Sept 28

Topcs: 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|.

Triangle's inequality: |a+b|<|a|+|b|

Solve: if you need more practice with sums, try the following:

calculate the sum in problem B1 4a; calculate 1-1/2+1/4-1/8+...+(-1)^n/2^n;

calculate 1+2+3+...+(8n+3); calculate 101+102+103+...+n (where n>100, for the sum to make sense).

Solve, 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

|x-1| < .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 not 1)?

5. Let us denote by epsilon a very, very small positive number (think like .00000000001, or like 10^(-10,000), or maybe even smaller...)

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|

Math tutor locations and hours

Friday Sept 28

Topcs: 2.5, A.2 Theorems 1, 2(proved), 3(proved), 4(proved i), 5.

Solve, 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) -> P(a)/Q(a) as 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
**