Text BS 4.1, 4.2,4.3, 5.1, 5.2, 5.3
P.1 Two mad flies f and g
P.1 Two mad flies f and g
Decided to collide against the wall (y -axis)!
P.1 Two mad flies f and g
Decided to collide against the wall (y -axis)!
Start at time t = 1; collision at t = 0.
P.1
Text BS 3.1, 3.2, 3.3, 3.4, 3.5, and a little of 3.6
P.1 What are sequences?
Df. A sequence of elements in S is a function f : N S.
P.1 What are sequences?
Df. A sequence of elements in S is a function f : N S. We view it as an
ordered list (a1 = f (1), a
Text GL 6.1,6.2,6.3,6.4,9.4,9.5,9.6
P.1 which one to take
Q A farmer had two sons and two pieces of lands.
He wants to give the bigger one to his elder son. But he does not know
which one is bigger! Can you help? Of course, yes! Use grids and count the
nu
Text BS 9.4
P.1 Power series, limsup
Df A power series around a is an expression P(x) =
P
n=0
an (x a)n .
Domain of convergence of P(x) is cfw_x | P(x) is convergent. 6= ?
Eg Each polynomial is a power series around 0. For us, it is enough to consider
pow
P.1 What are sequences?
Df. A sequence of elements in S is a function f : N S. We view it as an
ordered list (a1 = f (1), a2 = f (2), ).
Eg. a) The sequence 1, 2, 3, 4, 5 is written as (n).
1 1 1 1
b) ( 1 ) means 1, , , , ,
n
2 3 4 5
c) (an ) = (a,a,a,.)
Text BS 3.7, 9.1, 9.2, 9.3
Po. : proof outline.
Ex*: means for your information/ not for exam/ interested reader may
attempt and drop an email for hints.
P.1 Definition
Df A series is an expression a1 + a2 + a3 + , where
P.1 Definition
Df A series is an e
Text BS 6.1, 6.2,6.3, 6.4
P.1 Definitions
f (x)f (c)
exists, we call
xc
xc
d
dx f|x=c or Df (c). We say f
Df Let a < b, f : [a, b] R and c [a, b]. If lim
the derivative of f at c. Notation: f (c) or
differentiable at c if f
If c > a and lim
xc
(c)
it
is
P.1 Ordered Field
Df[ Ordered field]
A set F with two binary relations + and defined on it
satisfying these properties. Here a, b, c F and ab means a b.
F1[ closure]
a, b we have a + b, ab F.
F2[ associative]
a + (b + c ) = (a + b) + c , a(bc ) = (ab)c ho