The document discusses the p-series test and provides an example of using it to test the convergence of a series.
1) The p-series test states that the series Σ1/np converges if p>1 and diverges if p≤1.
2) As an example, it tests the series Σ(n+1)/(n+2)2 by comparing it to the divergent p-series Σ1/n, showing their limits are equal so the original series must also diverge.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
1) The document uses mathematical induction to prove several formulas.
2) It demonstrates proofs for formulas like 1 + 3 + 5 + ... + (2n-1) = n^2 and 2 + 4 + ... + 2n = n(n+1).
3) The proofs follow the standard structure of mathematical induction, showing the base case is true and using the induction hypothesis to show if the statement is true for n it is also true for n+1.
The document discusses integration by parts and the tableau method for evaluating integrals. It provides the formula for integration by parts, an example using the tableau method to evaluate the integral of (x3 + 2x)ex/2dx, and discusses how the tableau method can be used to evaluate integrals involving polynomials and functions that can be integrated repeatedly, such as ex, sin(x), and cos(x). It also provides the formulas for antiderivatives of trigonometric powers like cosn(x).
(1) The document discusses the gamma and beta functions. It provides definitions and key properties of each function.
(2) The gamma function is defined by an integral involving x, m, and e. Properties include relationships between gamma values when m is an integer or real number.
(3) The beta function is defined by an integral involving x, n, and m. It is also denoted as B(n,m). Properties relate the beta function to the gamma function and include formulas for specific values.
Mathematical Induction
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 18, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses the p-series test and provides an example of using it to test the convergence of a series.
1) The p-series test states that the series Σ1/np converges if p>1 and diverges if p≤1.
2) As an example, it tests the series Σ(n+1)/(n+2)2 by comparing it to the divergent p-series Σ1/n, showing their limits are equal so the original series must also diverge.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
1) The document uses mathematical induction to prove several formulas.
2) It demonstrates proofs for formulas like 1 + 3 + 5 + ... + (2n-1) = n^2 and 2 + 4 + ... + 2n = n(n+1).
3) The proofs follow the standard structure of mathematical induction, showing the base case is true and using the induction hypothesis to show if the statement is true for n it is also true for n+1.
The document discusses integration by parts and the tableau method for evaluating integrals. It provides the formula for integration by parts, an example using the tableau method to evaluate the integral of (x3 + 2x)ex/2dx, and discusses how the tableau method can be used to evaluate integrals involving polynomials and functions that can be integrated repeatedly, such as ex, sin(x), and cos(x). It also provides the formulas for antiderivatives of trigonometric powers like cosn(x).
(1) The document discusses the gamma and beta functions. It provides definitions and key properties of each function.
(2) The gamma function is defined by an integral involving x, m, and e. Properties include relationships between gamma values when m is an integer or real number.
(3) The beta function is defined by an integral involving x, n, and m. It is also denoted as B(n,m). Properties relate the beta function to the gamma function and include formulas for specific values.
Mathematical Induction
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 18, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
25 the ratio, root, and ratio comparison test xmath266
The document discusses two tests for determining if an infinite series converges or diverges: the ratio test and the root test. The ratio test checks if the limit of the ratio of successive terms is less than 1, equal to 1, or greater than 1, indicating convergence, inconclusive result, or divergence, respectively. The root test checks if the limit of the nth root of terms is less than, equal to, or greater than 1, with similar implications for convergence or divergence. Examples are provided to demonstrate how to apply each test. It is also noted that if a series cannot be directly tested, an equivalent surrogate series may be used instead.
The document discusses basic concepts related to continuous functions. It begins with an introduction and motivation for studying continuous functions. Some key reasons mentioned are that continuous functions are needed for integration and as underlying functions in differential equations. The document then provides definitions of limits and continuity in terms of limits. It gives examples of determining limits and continuity for various functions. Contributors to the field like Bolzano, Cauchy, and Weierstrass are also acknowledged. The document concludes with additional definitions of continuity, examples, and discussions of uniform continuity.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
This document is the preface and contents for a set of notes on complex analysis. It was prepared by Charudatt Kadolkar for MSc students at IIT Guwahati in 2000 and 2001. The notes cover topics such as complex numbers, functions of complex variables, analytic functions, integrals, series, and the theory of residues and its applications. The contents section provides an outline of the chapters and topics to be covered.
The document discusses convergence of sequences and power series. It defines convergence of a sequence and states that the limit of a convergent sequence is unique. It also discusses Taylor series and Laurent series, stating that if a function f(z) is analytic inside a circle C with center z0, its Taylor series representation about z0 will converge to f(z) for all z inside C. Similarly, if f(z) is analytic in an annular region bounded by two concentric circles, its Laurent series will represent f(z) in that region.
This document discusses Taylor series expansions. It defines Taylor series as the expansion of a complex function f(z) that is analytic inside and on a simple closed curve C in the z-plane. The Taylor series expresses f(z) as a power series centered at a point z0 within C. It provides examples of standard Taylor series expansions and worked illustrations of expanding various functions as Taylor series. The document also notes that the radius of convergence of a Taylor series is defined by the distance to the nearest singularity from the center point z0.
24 the harmonic series and the integral test xmath266
The document discusses the harmonic series, which is the series of terms 1/n as n goes from 1 to infinity. While the individual terms of this series approach 0 as n increases, the total sum of the terms is actually infinite. This demonstrates that a series can converge to 0 without the overall sum converging. The document proves this by summing the harmonic series terms in blocks, showing that each block sum is greater than 9/10, so the total sum must be infinite.
The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.
This document discusses relations and functions for class 12. It defines concepts like cartesian product, relations, domain and codomain, range, types of relations such as reflexive, symmetric, transitive and equivalence relations. It also defines functions, one-to-one and onto functions, bijective functions, composition of functions, and inverse functions. An example is provided to show that a given relation is reflexive but not symmetric or transitive. Another example shows a given function is one-to-one but not onto.
Any analytic function is locally represented by a convergent power series and is infinitely differentiable. Real analytic functions are defined on an open set of the real line, while complex analytic functions are defined on an open set of the complex plane. Both are infinitely differentiable, but complex analytic functions have additional properties like Liouville's theorem stating bounded complex analytic functions defined on the whole complex plane are constant. Real analytic functions do not have this property and their power series need only converge locally rather than on the entire domain.
This document introduces various concepts and methods related to mathematical proofs. It defines key terminology like theorems, propositions, lemmas, corollaries, and conjectures. It also describes different types of proofs like direct proofs, proofs by contraposition, and proofs of equivalence. Examples are provided to illustrate direct proofs for statements about odd integers and perfect squares.
35 tangent and arc length in polar coordinatesmath266
The document discusses parametric representations of polar curves. It begins by explaining that a rectangular curve given by y=f(x) can be parametrized as x=t and y=f(t). It then explains that a polar curve given by r=f(θ) can be parametrized as x=f(θ)cos(θ) and y=f(θ)sin(θ). Examples are given of parametrizing the Archimedean spiral r=θ and the cardioid r=1+sin(θ). Formulas are derived for calculating the slope of the tangent line to a polar curve at a given point in terms of r and θ. Examples are worked out for finding the slope
This document discusses logical expressions and quantifiers. It provides examples of using quantifiers like ∀ and ∃ to represent statements involving sets, predicates and relationships between variables. It also discusses the meanings of multiple quantifiers, bound and free variables, and how to translate statements into logical expressions.
The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Hyperbolic functions are useful in mathematics and physics. The main hyperbolic functions are the hyperbolic sine, cosine, and tangent. Hyperbolic curves include the catenary curve, which describes the shape of a hanging chain and is modeled by the hyperbolic cosine function. Hyperbolic functions are applied in areas like physics, differential equations, and special relativity.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
25 the ratio, root, and ratio comparison test xmath266
The document discusses two tests for determining if an infinite series converges or diverges: the ratio test and the root test. The ratio test checks if the limit of the ratio of successive terms is less than 1, equal to 1, or greater than 1, indicating convergence, inconclusive result, or divergence, respectively. The root test checks if the limit of the nth root of terms is less than, equal to, or greater than 1, with similar implications for convergence or divergence. Examples are provided to demonstrate how to apply each test. It is also noted that if a series cannot be directly tested, an equivalent surrogate series may be used instead.
The document discusses basic concepts related to continuous functions. It begins with an introduction and motivation for studying continuous functions. Some key reasons mentioned are that continuous functions are needed for integration and as underlying functions in differential equations. The document then provides definitions of limits and continuity in terms of limits. It gives examples of determining limits and continuity for various functions. Contributors to the field like Bolzano, Cauchy, and Weierstrass are also acknowledged. The document concludes with additional definitions of continuity, examples, and discussions of uniform continuity.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
This document is the preface and contents for a set of notes on complex analysis. It was prepared by Charudatt Kadolkar for MSc students at IIT Guwahati in 2000 and 2001. The notes cover topics such as complex numbers, functions of complex variables, analytic functions, integrals, series, and the theory of residues and its applications. The contents section provides an outline of the chapters and topics to be covered.
The document discusses convergence of sequences and power series. It defines convergence of a sequence and states that the limit of a convergent sequence is unique. It also discusses Taylor series and Laurent series, stating that if a function f(z) is analytic inside a circle C with center z0, its Taylor series representation about z0 will converge to f(z) for all z inside C. Similarly, if f(z) is analytic in an annular region bounded by two concentric circles, its Laurent series will represent f(z) in that region.
This document discusses Taylor series expansions. It defines Taylor series as the expansion of a complex function f(z) that is analytic inside and on a simple closed curve C in the z-plane. The Taylor series expresses f(z) as a power series centered at a point z0 within C. It provides examples of standard Taylor series expansions and worked illustrations of expanding various functions as Taylor series. The document also notes that the radius of convergence of a Taylor series is defined by the distance to the nearest singularity from the center point z0.
24 the harmonic series and the integral test xmath266
The document discusses the harmonic series, which is the series of terms 1/n as n goes from 1 to infinity. While the individual terms of this series approach 0 as n increases, the total sum of the terms is actually infinite. This demonstrates that a series can converge to 0 without the overall sum converging. The document proves this by summing the harmonic series terms in blocks, showing that each block sum is greater than 9/10, so the total sum must be infinite.
The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.
This document discusses relations and functions for class 12. It defines concepts like cartesian product, relations, domain and codomain, range, types of relations such as reflexive, symmetric, transitive and equivalence relations. It also defines functions, one-to-one and onto functions, bijective functions, composition of functions, and inverse functions. An example is provided to show that a given relation is reflexive but not symmetric or transitive. Another example shows a given function is one-to-one but not onto.
Any analytic function is locally represented by a convergent power series and is infinitely differentiable. Real analytic functions are defined on an open set of the real line, while complex analytic functions are defined on an open set of the complex plane. Both are infinitely differentiable, but complex analytic functions have additional properties like Liouville's theorem stating bounded complex analytic functions defined on the whole complex plane are constant. Real analytic functions do not have this property and their power series need only converge locally rather than on the entire domain.
This document introduces various concepts and methods related to mathematical proofs. It defines key terminology like theorems, propositions, lemmas, corollaries, and conjectures. It also describes different types of proofs like direct proofs, proofs by contraposition, and proofs of equivalence. Examples are provided to illustrate direct proofs for statements about odd integers and perfect squares.
35 tangent and arc length in polar coordinatesmath266
The document discusses parametric representations of polar curves. It begins by explaining that a rectangular curve given by y=f(x) can be parametrized as x=t and y=f(t). It then explains that a polar curve given by r=f(θ) can be parametrized as x=f(θ)cos(θ) and y=f(θ)sin(θ). Examples are given of parametrizing the Archimedean spiral r=θ and the cardioid r=1+sin(θ). Formulas are derived for calculating the slope of the tangent line to a polar curve at a given point in terms of r and θ. Examples are worked out for finding the slope
This document discusses logical expressions and quantifiers. It provides examples of using quantifiers like ∀ and ∃ to represent statements involving sets, predicates and relationships between variables. It also discusses the meanings of multiple quantifiers, bound and free variables, and how to translate statements into logical expressions.
The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
Hyperbolic functions are useful in mathematics and physics. The main hyperbolic functions are the hyperbolic sine, cosine, and tangent. Hyperbolic curves include the catenary curve, which describes the shape of a hanging chain and is modeled by the hyperbolic cosine function. Hyperbolic functions are applied in areas like physics, differential equations, and special relativity.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
The document discusses strategies for lifelong learning in a changing work environment. It states that to be successful in the next decade, individuals will need to demonstrate foresight in navigating changing skills requirements and continually reassess and update their skills. Workers will need to be adaptable lifelong learners. The document emphasizes the importance of continual learning.
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