On exercise #59 in Section 2
... value for a that might cause the limit to exist. But here is a way to support that reasoning. We know that there would be no hope of having a limit exist, unless the numerator did approach zero as x approaches -2, because if the numerator approached any non-zero value, while the denominator was appr ...
... value for a that might cause the limit to exist. But here is a way to support that reasoning. We know that there would be no hope of having a limit exist, unless the numerator did approach zero as x approaches -2, because if the numerator approached any non-zero value, while the denominator was appr ...
IGCSE(E) PPQs on Sequences
... The first three diagrams in a sequence are shown above. The diagrams are made up of dots and lines. Each line is one centimetre long. ...
... The first three diagrams in a sequence are shown above. The diagrams are made up of dots and lines. Each line is one centimetre long. ...
Exercises Warm Up to the Theory of Computation
... ------------------------------------8. Prove that the square root of 8 is irrational. 9. Prove or disprove the following statements: (a) The sum of a rational and an irrational number must be irrational. (b) The sum of two positive irrational numbers must be irrational. (c) The product of two irrati ...
... ------------------------------------8. Prove that the square root of 8 is irrational. 9. Prove or disprove the following statements: (a) The sum of a rational and an irrational number must be irrational. (b) The sum of two positive irrational numbers must be irrational. (c) The product of two irrati ...
13.2 Explicit Sequences
... What are two types of Sequences? • Arithmetic Sequence: Is a sequence in which each term is equal to the previous term plus a constant. This constant is called the common difference. ...
... What are two types of Sequences? • Arithmetic Sequence: Is a sequence in which each term is equal to the previous term plus a constant. This constant is called the common difference. ...
Solution
... (a) The set P(N>0 ), the power set of the positive natural numbers. (b) The set S of all functions f : {0, 1} → N>0 . (c) The set T of all functions f : N>0 → {0, 1}. (d) The set A × A, where A is an infinite countable set. Solution: (a) The set P(N>0 ) is uncountable because the power set of a set ...
... (a) The set P(N>0 ), the power set of the positive natural numbers. (b) The set S of all functions f : {0, 1} → N>0 . (c) The set T of all functions f : N>0 → {0, 1}. (d) The set A × A, where A is an infinite countable set. Solution: (a) The set P(N>0 ) is uncountable because the power set of a set ...
THE DIVISOR PROBLEM ON SQUARE
... Could the bound be reduced? The answer is definite. Hence we have studied further more the asymptotic properties of and have got some non-trivial properties about it. About this problem, we know very little at present. At least we have not found it in any reference that we could find. Therefore, in ...
... Could the bound be reduced? The answer is definite. Hence we have studied further more the asymptotic properties of and have got some non-trivial properties about it. About this problem, we know very little at present. At least we have not found it in any reference that we could find. Therefore, in ...
Notes/summary
... Trailing 5’s squaring trick: to find the square of a two-digit number ending in 5, take the tens digit, add one, and multiply this by the original tens digit. Then append 25 to the end of the result. Example: to find 452, take the tens digit, 4. Add one to get 5. Multiply it by the original tens dig ...
... Trailing 5’s squaring trick: to find the square of a two-digit number ending in 5, take the tens digit, add one, and multiply this by the original tens digit. Then append 25 to the end of the result. Example: to find 452, take the tens digit, 4. Add one to get 5. Multiply it by the original tens dig ...
DEFINITION OF A SEQUENCE
... If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent. A sequence can converge to only one limit, i.e., if a limit exists, it is unique. See Problem 2.8. A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequenc ...
... If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent. A sequence can converge to only one limit, i.e., if a limit exists, it is unique. See Problem 2.8. A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequenc ...
Extra Problem Set I Countable and Uncountable Sets
... where the digits ai belong to {0, 1, . . . , 9}. For example 1/2 is 0.5 and so a1 = 5, a2 = 0, a3 = 0 etc. Another example, 1/3 is 0.33 . . . and so a1 = 3, a2 = 3, a3 = 3, etc. Write an argument which explains why this decimal expansion representation of a number is almost unique. The only ambiguit ...
... where the digits ai belong to {0, 1, . . . , 9}. For example 1/2 is 0.5 and so a1 = 5, a2 = 0, a3 = 0 etc. Another example, 1/3 is 0.33 . . . and so a1 = 3, a2 = 3, a3 = 3, etc. Write an argument which explains why this decimal expansion representation of a number is almost unique. The only ambiguit ...
Geometry (H) Lesson 2.1 2.1 Notes: Inductive Reasoning Lesson
... Part I: Practice 1. Find a pattern for each sequence. Use the pattern to show the next two terms. ...
... Part I: Practice 1. Find a pattern for each sequence. Use the pattern to show the next two terms. ...
Math 131 notes
... • Note that each successive column grows just as quickly as the first. • Divide the first column by 4, second by 8, etc., and what happens? The columns are the same. • Successive differences of a geometric sequence still are geometric sequences. ...
... • Note that each successive column grows just as quickly as the first. • Divide the first column by 4, second by 8, etc., and what happens? The columns are the same. • Successive differences of a geometric sequence still are geometric sequences. ...