Problem Set 3
... Unlike the previous problem set, in this one you will need to prove your claims rigorously. 1. (a) Prove Bernoulli’s inequality: (1 + x)n ≥ 1 + nx for every real number x ≥ −1 and every n ∈ N. (b) Define the sequence (an )n∈N and (bn )n∈N by ...
... Unlike the previous problem set, in this one you will need to prove your claims rigorously. 1. (a) Prove Bernoulli’s inequality: (1 + x)n ≥ 1 + nx for every real number x ≥ −1 and every n ∈ N. (b) Define the sequence (an )n∈N and (bn )n∈N by ...
Situation 21: Exponential Rules
... When the domain of x is extended to the negative real numbers or 0, then the domain of m is limited to those values for which xm is defined. A more general principle is that extending the domain of a function of two variables might mean restricting the domain of one of the variables. If x m is to be ...
... When the domain of x is extended to the negative real numbers or 0, then the domain of m is limited to those values for which xm is defined. A more general principle is that extending the domain of a function of two variables might mean restricting the domain of one of the variables. If x m is to be ...
Full text
... equations Fn^k. Thus, if A is palindromic, then the equations Fntk, for k = 1, 2,...,«, hold for infinitely many n. D To see how a positive irrational number a can be used to generate palindromic sequences, we recall certain customary notations from the theory of continued fractions. Suppose a has a ...
... equations Fn^k. Thus, if A is palindromic, then the equations Fntk, for k = 1, 2,...,«, hold for infinitely many n. D To see how a positive irrational number a can be used to generate palindromic sequences, we recall certain customary notations from the theory of continued fractions. Suppose a has a ...
Rational Numbers
... • Repeating decimals can always be written as fractions, so repeating decimals are always rational numbers. You can use bar notation to indicate that some part of a decimal repeats forever, for example, 0.333… 0.3 ...
... • Repeating decimals can always be written as fractions, so repeating decimals are always rational numbers. You can use bar notation to indicate that some part of a decimal repeats forever, for example, 0.333… 0.3 ...
s01.pdf
... Besides rounding errors, the two other types of errors often encounterd in numerical computing are discretization errors (resulting when a continuous problem is replaced by its discrete analogue) and convergence errors (resulting from the termination of an in nite sequence after a nite number of te ...
... Besides rounding errors, the two other types of errors often encounterd in numerical computing are discretization errors (resulting when a continuous problem is replaced by its discrete analogue) and convergence errors (resulting from the termination of an in nite sequence after a nite number of te ...
Number Patterns: Introduction
... In earlier grades you saw patterns in the form of pictures and numbers. In this chapter, we learn more about the mathematics of patterns. Patterns are recognisable as repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, ...
... In earlier grades you saw patterns in the form of pictures and numbers. In this chapter, we learn more about the mathematics of patterns. Patterns are recognisable as repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, ...
Descartes`s Rule of Signs & Bounds: Things that make your life easier
... 1. The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer. 2. The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer ...
... 1. The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer. 2. The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer ...
