infinite series
... the first 100 terms in (2) is approximately 3.15, which isn’t even right in the second digit after the decimal point. The reason (2) is slowly convergent is that it comes from evaluating the infinite series for arctan t at an endpoint of its interval of convergence, and power series tend to converge ...
... the first 100 terms in (2) is approximately 3.15, which isn’t even right in the second digit after the decimal point. The reason (2) is slowly convergent is that it comes from evaluating the infinite series for arctan t at an endpoint of its interval of convergence, and power series tend to converge ...
these notes by Samir Siksek from a Warwick university 1st year course.
... Your lectures are excruciatingly boring. Besides, 12 noon is a perversely early time to schedule a lecture and no self-respecting student can be expected to be awake yet. Do I really have to attend your lectures, or can I make do with these lecture notes? I’ll take that as an endorsement of the grea ...
... Your lectures are excruciatingly boring. Besides, 12 noon is a perversely early time to schedule a lecture and no self-respecting student can be expected to be awake yet. Do I really have to attend your lectures, or can I make do with these lecture notes? I’ll take that as an endorsement of the grea ...
Chapter 3 - Molecular Symmetry
... Rx, Ry, Rz stand for rotations about the specified axes. Symbols in the column on the far left are Mulliken Labels. ...
... Rx, Ry, Rz stand for rotations about the specified axes. Symbols in the column on the far left are Mulliken Labels. ...
Output - UVa Online Judge
... Mahbub thinks that he has found something interesting but he is not sure whether he is right or not. For each integer he seems to find a multiple of it such that it is only composed of 3s and 0s. Can you help him? You will be given a positive integer K. You have to find a positive multiple of K whic ...
... Mahbub thinks that he has found something interesting but he is not sure whether he is right or not. For each integer he seems to find a multiple of it such that it is only composed of 3s and 0s. Can you help him? You will be given a positive integer K. You have to find a positive multiple of K whic ...
Representing Numbers Using Fibonacci Variants
... Existence by induction: 1 = f2 , 2 = f3 , 3 = f4 . Assume every integer from 1 to n has a Zeckendorf representation. If n + 1 is a Fibonacci number, done. Otherwise, there is some j such that fj < n + 1 < fj+1 . Now n + 1 − fj < n, so has a Zeckendorf representation, and n + 1 − fj < fj+1 − fj = fj− ...
... Existence by induction: 1 = f2 , 2 = f3 , 3 = f4 . Assume every integer from 1 to n has a Zeckendorf representation. If n + 1 is a Fibonacci number, done. Otherwise, there is some j such that fj < n + 1 < fj+1 . Now n + 1 − fj < n, so has a Zeckendorf representation, and n + 1 − fj < fj+1 − fj = fj− ...
Section 4.1 - Montgomery College
... An expression involving exponents is simplified if 1. It includes no parentheses. 2. Each variable or constant appears as a base as few times as possible. For example, we write x2x4 = x6 3. Each numerical expression (such as 72) has been calculated, and each numerical fraction has been simplified. 4 ...
... An expression involving exponents is simplified if 1. It includes no parentheses. 2. Each variable or constant appears as a base as few times as possible. For example, we write x2x4 = x6 3. Each numerical expression (such as 72) has been calculated, and each numerical fraction has been simplified. 4 ...
0 q w e r +u +u +u +u +u __ __ __ __ __ t y u i o +u +u +u +u +u __
... Directions: No guessing! If you know the answer to each problem, write it down. If not, count by the divisor up to the dividend making a tally mark in the box as you say each number. Count the tally marks and write the answer. Say the problem and answer quietly. ...
... Directions: No guessing! If you know the answer to each problem, write it down. If not, count by the divisor up to the dividend making a tally mark in the box as you say each number. Count the tally marks and write the answer. Say the problem and answer quietly. ...
A Guide to Fractions
... (N.B Whatever you do to the numerator you must do to the denominator!) Equivalent fractions are fractions which have the same value. They are equal. You can make equivalent fractions by multiplying or dividing both the numerator and denominator by the same amount. You only multiply or divide, never ...
... (N.B Whatever you do to the numerator you must do to the denominator!) Equivalent fractions are fractions which have the same value. They are equal. You can make equivalent fractions by multiplying or dividing both the numerator and denominator by the same amount. You only multiply or divide, never ...
Addition
Addition (often signified by the plus symbol ""+"") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together; making a total of 5 apples. This observation is equivalent to the mathematical expression ""3 + 2 = 5"" i.e., ""3 add 2 is equal to 5"".Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers, rational numbers, real numbers and complex numbers and other abstract objects such as vectors and matrices.In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly.Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some non-human animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.