Series

Def Series Convergence Let ${a_{j}} \in R^{n}$ and $S_{n} = \sum_{j = 1}^{n} a_{j}$ . We say that the terms $S_{n}$ are the partial sums of the series $\sum_{j = 1}^{\infty} a_{j}$ . We say the series $\sum_{j = 1}^{\infty} a_{j}$ converges if $lim_{n \to \infty} S_{n}$ exists, and diverges otherwise.

Lecture 13: The Riemann Integral

This lecture introduces the concept of the

Riemann integral69.

A

partition P of an interval [a,b] is a sequence of points a=t0<t1<⋯<tn=b70.
The

lower area for a bounded function f with respect to a partition P is L(f,P)=∑i=1nmi(ti−ti−1), where mi is the infimum of the function on the subinterval [ti−1,ti]71.
The

upper area is U(f,P)=∑i=1nMi(ti−ti−1), where Mi is the supremum72.

A function

f is integrable on [a,b] if the supremum of the lower areas equals the infimum of the upper areas for all possible partitions73. This common value is the definite integral, denoted

∫abf(x)dx74. The lecture notes mention that the Dirichlet function (a function that is 1 for rational numbers and 0 for irrational numbers) is not integrable75.

Lecture 14: Uniform Continuity

This lecture presents the theorem that a

**continuous function on a closed interval [a,b] is integrable on [a,b]**76. The proof relies on a stronger form of continuity called

uniform continuity77. A function

f is uniformly continuous on an interval I if for every ϵ>0, there exists a δ>0 such that for all x,y∈I, if ∣x−y∣<δ, then ∣f(x)−f(y)∣[citestart]<ϵ78. The key difference from regular continuity is that

δ depends only on ϵ, not on the specific point79.

The lecture states that a continuous function on a closed interval

[a,b] is also uniformly continuous on that interval80. This property is then used to show that for any

ϵ>0, a partition P can be found such that the difference between the upper and lower areas, U(f,P)−L(f,P), is less than ϵ, which proves integrability81.

Lecture 15: The Fundamental Theorem of Calculus

This lecture introduces the

Fundamental Theorem of Calculus (FTC)82. The notes show that for a continuous function

f on [a,b], if we define F(x)=∫axf(t)dt, then F is a continuous function, and if f is continuous at c, then F is differentiable at c and F′(c)=f(c)83.

The lecture also restates the

Mean Value Theorem for Integrals, which states that if f is integrable on [a,b], there exists a point c in [a,b] such that ∫abf(x)dx=f(c)(b−a)84.

Lecture 16: Hydrostatic Pressure and Integration

This lecture applies integration to calculate hydrostatic pressure85. The total force is the integral of the pressure over the area86. This lecture also introduces the idea of defining a function based on a given property, rather than by a formula87.

Lecture 17: Trigonometric Functions

This lecture defines sine and cosine functions in terms of the integral of a circle88. The notes also mention that cosine and sine can be extended to all real numbers, and their properties (e.g., periodicity) can be derived from these integral definitions89.

Lecture 18: Logarithmic and Exponential Functions

The natural logarithm, log(x), is defined as an integral:

log(x)=∫1xt1dt

for

x>09090. The exponential function,

exp(x), is defined as the inverse of the logarithm, and the number e is defined as exp(1)91. The lecture notes prove that

log(xy)=log(x)+log(y) and that exp(x+y)=exp(x)exp(y)92929292.

Lecture 19: Integration by Parts and Substitution

This lecture connects differentiation rules to integration techniques93.

Product Rule leads to Integration by Parts: ∫udv=uv−∫vdu94. An example is provided for

∫xexdx95.
Chain Rule leads to Integration by Substitution96.

Lecture 20: Polynomial Approximation

This lecture introduces the concept of

polynomial approximation97. The notes define a “good approximation” as a function

L(x) where limx→ax−af(x)−L(x)=098. The lecture states that a function has a good linear approximation at a point

a if and only if it is differentiable at a, and the best linear approximation is given by the tangent line, L(x)=f′(a)(x−a)+f(a)99.

The lecture then introduces

Taylor polynomials, which are polynomials of degree n that have the same derivatives as the function at a point a up to the nth order100. The general form of the

nth degree Taylor polynomial of f at a is:

Pn,a(x)=k=0∑nk!f(k)(a)(x−a)k

The notes state that the limit of the difference between the function and its Taylor polynomial, divided by

(x−a)n, as x→a is 0101. An example is provided for finding the Taylor polynomial for

ex at a=0102.

Lecture 21: Taylor’s Theorem with Remainder

This lecture presents

Taylor’s Theorem, which provides a formula for the remainder term, Rn,a(x)=f(x)−Pn,a(x)103. The theorem states that if the

(n+1)th derivative of f is continuous on [a,x], then there exists a number c in (a,x) such that the remainder is given by:

Rn,a(x)=(n+1)!f(n+1)(c)(x−a)n+1

This theorem allows for an estimation of the error in the Taylor polynomial approximation104. The proof is a sophisticated application of the Mean Value Theorem105.

Lecture 22: P vs NP

This lecture provides an overview of the

P vs NP problem, a major unsolved problem in computer science106.

P (Polynomial time) is the class of decision problems for which a solution can be found in a time that is a polynomial function of the input size107. Examples include calculating the greatest common divisor, determining if an integer is prime, and string matching108.
NP (Nondeterministic Polynomial time) is the class of decision problems for which a given solution can be verified in polynomial time109. Examples include the Traveling Salesman Problem, Sudoku, and graph coloring110.

The lecture states that the relationship between P and NP is a Millennium Prize Problem with a $1,000,000 reward111. The core question is whether every problem whose solution can be quickly verified can also be quickly solved112.

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