Mathematical Analysis: Sequences and Series

Study Notes

Mathematical analysis is a branch of mathematics that deals with the study of limits, convergence, and continuity of functions and sequences.
It provides a rigorous foundation for calculus and is used to develop the theory of calculus.

Sequences: A sequence is an ordered list of numbers, denoted by {an} or (an). It converges to a limit L if for any ε > 0, there exists a natural number N such that for all n ≥ N, |an - L| < ε.
Series: A series is the sum of a sequence of terms. It converges if the sequence of partial sums converges.
Continuity: A function f is continuous at a point x=a if for any ε > 0, there exists a δ > 0 such that for all x, |x-a| < δ implies |f(x) - f(a)| < ε.
Differentiability: A function f is differentiable at a point x=a if the limit lim(x→a) [f(x) - f(a)]/[x - a] exists.

Bolzano-Weierstrass Theorem: Every bounded sequence in ℝ has a convergent subsequence.
Heine-Borel Theorem: A subset of ℝ is compact if and only if it is closed and bounded.
Intermediate Value Theorem: If a function f is continuous on [a, b] and k is between f(a) and f(b), then there exists a c in [a, b] such that f(c) = k.

Calculus: Mathematical analysis provides a rigorous foundation for calculus, including the study of limits, derivatives, and integrals.
Real Analysis: Mathematical analysis is used to study the properties of real-valued functions and sequences.
Functional Analysis: Mathematical analysis is used to study the properties of vector spaces and linear operators.

Augustin-Louis Cauchy: Developed the theory of calculus and introduced the concept of convergence.
Karl Weierstrass: Developed the concept of epsilon-delta definition of limit and introduced the concept of uniform convergence.
Richard Courant: Developed the theory of calculus and introduced the concept of functional analysis.

Mathematical analysis is a branch of mathematics that deals with the study of limits, convergence, and continuity of functions and sequences.
It provides a rigorous foundation for calculus and is used to develop the theory of calculus.

Sequences: An ordered list of numbers, denoted by {an} or (an), that converges to a limit L if for any ε > 0, there exists a natural number N such that for all n ≥ N, |an - L| < ε.
Series: The sum of a sequence of terms that converges if the sequence of partial sums converges.
Continuity: A function f is continuous at a point x=a if for any ε > 0, there exists a δ > 0 such that for all x, |x-a| < δ implies |f(x) - f(a)| < ε.
Differentiability: A function f is differentiable at a point x=a if the limit lim(x→a) [f(x) - f(a)]/[x - a] exists.

Bolzano-Weierstrass Theorem: Every bounded sequence in ℝ has a convergent subsequence.
Heine-Borel Theorem: A subset of ℝ is compact if and only if it is closed and bounded.
Intermediate Value Theorem: If a function f is continuous on [a, b] and k is between f(a) and f(b), then there exists a c in [a, b] such that f(c) = k.

Calculus: Mathematical analysis provides a rigorous foundation for calculus, including the study of limits, derivatives, and integrals.
Real Analysis: Mathematical analysis is used to study the properties of real-valued functions and sequences.
Functional Analysis: Mathematical analysis is used to study the properties of vector spaces and linear operators.

Augustin-Louis Cauchy: Developed the theory of calculus and introduced the concept of convergence.
Karl Weierstrass: Developed the concept of epsilon-delta definition of limit and introduced the concept of uniform convergence.
Richard Courant: Developed the theory of calculus and introduced the concept of functional analysis.