MA 101 (Mathematics I)
Sequence : Summary of Lectures
A sequence of real numbers or a sequence in R is a mapping f : N R.
Notation: We write xn for f (n), n N and so the notation for a sequence is (xn ).
Examples:
1. Constant sequence: (a, a, a, .), where
Riemann Integral: Motivation
Riemann Integral: Motivation
Partition of [a, b]: A nite set cfw_x0 , x1 , ., xn [a, b] such that
a = x0 < x1 < < xn = b.
Riemann Integral: Motivation
Partition of [a, b]: A nite set cfw_x0 , x1 , ., xn [a, b] such that
a =
An innite series in R is an expression
n=1
where (xn ) is a sequence in R.
xn ,
An innite series in R is an expression
xn ,
n=1
where (xn ) is a sequence in R.
More formally, it is an ordered pair (xn ), (sn ),
where (xn ) is a sequence in R,
and sn = x1
A sequence of real numbers or a sequence in R
is a mapping f : N R.
A sequence of real numbers or a sequence in R
is a mapping f : N R.
Notation: We write xn for f (n), n N
and so the notation for a sequence is (xn ).
A sequence of real numbers or a seque
Differentiability and Derivative: Let D R and let x0 D such
that there exists an interval I of R satisfying x0 I D.
Differentiability and Derivative: Let D R and let x0 D such
that there exists an interval I of R satisfying x0 I D.
A function f : D R is s
MA 101 (Mathematics - I)
Innite Series : Summary of Lectures
An innite series in R is an expression
xn , where (xn ) is a sequence in R.
n=1
More formally, it is an ordered pair (xn ), (sn ), where (xn ) is a sequence in R and
sn = x1 + + xn for all n N.
MA 101 (Mathematics I)
Integration : Summary of Lectures
Riemann Integral: Motivation
Partition of [a, b]: A nite set cfw_x0 , x1 , ., xn [a, b] such that a = x0 < x1 < < xn = b.
Upper sum & Lower sum: Let f : [a, b] R be bounded. For a partition P = cfw
Denition: Let D(= ) R and let f : D R.
Denition: Let D(= ) R and let f : D R.
We say that f is continuous at x0 D if for each > 0, there
exists > 0 such that |f (x) f (x0 )| < for all x D satisfying
|x x0 | < .
Denition: Let D(= ) R and let f : D R.
We sa
MA 101 (Mathematics I)
Dierentiation : Summary of Lectures
Dierentiability and Derivative: Let D R and let x0 D such that there exists
an interval I of R satisfying x0 I D.
(x
A function f : D R is said to be dierentiable at x0 if lim f (x)f 0 0 ) (or equ