Posts Tagged ‘sequence’
Thursday, December 10th, 2009
Overview of the Bounded Monotone Sequence Theorem
Description
A detailed tutorial on the bounded monotone sequence theorem. Step by step tutorial including several examples of the bounded monotone sequence theorem for reference.
Overview
The bounded monotone sequence theorem actually has several parts to it. First, you need to find out if something is bounded above or bounded below. The sequence is bounded above if there exists a real number B such that x sub n is less than or equal to B. The sequence is bounded below if there exists a real number B such that x sub n is greater than or equal to B. If something is a bounded sequence, that means it is bounded both above and below. Absolute values are also very important in determining the bounded sequence. The bounded monotone sequence theorem states that for every bounded monotone sequence x, there is a real number L such that x sub n implies L.
Tags: above, absolute, algebra, below, bounded, boundedness, equal to, greater than, implies, less than, monotone, number, real, sequence, theorem, value
Posted in Algebra | No Comments »
Thursday, November 19th, 2009
How to Find the Common Ratio of a Geometric Series
Description
A detailed tutorial on how to find the common ratio of a geometric series. Step by step tutorial including several examples of the common ratio for reference.
Overview
The common ratio is part of a geometric series, used commonly in calculus. The common ratio is the ratio of each term to the next – in other words, the common ratio is the pattern that the series or sequence follows. This is possible because in a geometric series, terms are only being multiplied by one number to get the next number, and it is always the same number. If a series is not geometric, it will not have a common ratio.
Tags: Calculus, common, geometric, multiplication, multiply, number, pattern, ratio, sequence, series, term
Posted in Calculus | No Comments »
Thursday, November 12th, 2009
How to Find the Next Term in an Arithmetic Sequence
Description
A detailed tutorial on finding the next term of an arithmetic sequence. Step by step tutorial including several examples of arithmetic sequences for reference.
Overview
Arithmetic sequences are sequences of numbers that are written in a particular pattern. Most problems including an arithmetic sequence don’t include all the terms in the sequence, and you have to find the next one in the sequence. In order to do this, you must find the pattern. The pattern can be anything – the same number could be added, subtracted, mutliplied, or divided from each previous number of the sequence. The previous number could be added to the number after it to come up with the next number. Each number in the sequence could be divisible by the same number. All numbers could be perfect or prime. There are an endless number of patterns, all you have to do is look and then follow that pattern to come up with the next term or terms in the sequence.
Tags: add, arithmetic, divide, mutliply, next, number, pattern, perfect, previous, prime, sequence, subtract, term
Posted in Arithmetic | No Comments »
Thursday, November 5th, 2009
Overview of the Monotone Convergence Theorem
Description
A detailed tutorial on the monotone convergence theorem. Step by step tutorial including several examples of the monotone convergence theorem for reference.
Overview
There are several different theorems that the term “monotone convergence” can apply to. However, the most important one, and the one most common called the monotone convergence theorem, is the Lebesgue Monotone Convergence Theorem. This particular monotone convergence theorem deals with calculus, and with integrals and limits specifically. It is a more general form of the other two monotone convergence theorems, which is why it is considered to be the most important.
Tags: Calculus, converge, convergence, form, general, integral, Lebesgue, limit, monotone, number, real, sequence, series, theorem
Posted in Calculus | No Comments »
Thursday, November 5th, 2009
Introduction to Nested Intervals
Description
A detailed tutorial on nested intervals and the nested interval theorem. Step by step tutorial including several examples of nested intervals for reference.
Overview
Nested intervals means to have one interval (or multiple intervals) inside of another interval. The intervals will get smaller and smaller the more you add, until they will finally dimish entirely. There is a theorem for nested intervals, called the nested interval theorem. It states that if A_n = [a_n, b_n] is a sequence of closed intervals such that A_n+1 is a subset of A_n for all n belonging to the set of natural numbers, then the union over A_n is not an empty set.
Tags: algebra, closed, empty, interval, natural, nested, number, open, sequence, set, subset, theorem
Posted in Algebra | No Comments »
Friday, October 30th, 2009
Overview of Summation by Parts
Description
A detailed tutorial on summation by parts. Step by step tutorial including several examples of summation by parts for reference.
Overview
Summation by parts transforms the summation of products of sequences into other summations. Often it will simplify the computation of certain sums. Summation by parts is also referred to as Abel’s lemma or Abel’s transformation. Summation by parts is similar to integration by parts, only by using summation instead of integration. In mathematical notation, summation by parts can be written as: ![\sum_{k=m}^n f_k(g_{k+1}-g_k) = \left[f_{n+1}g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}(f_{k+1}- f_k)](http://s.wordpress.com/latex.php?latex=%5Csum_%7Bk%3Dm%7D%5En%20f_k%28g_%7Bk%2B1%7D-g_k%29%20%3D%20%5Cleft%5Bf_%7Bn%2B1%7Dg_%7Bn%2B1%7D%20-%20f_m%20g_m%5Cright%5D%20-%20%5Csum_%7Bk%3Dm%7D%5En%20g_%7Bk%2B1%7D%28f_%7Bk%2B1%7D-%20f_k%29&bg=ffffff&fg=000000&s=0 "\sum_{k=m}^n f_k(g_{k+1}-g_k) = \left[f_{n+1}g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}(f_{k+1}- f_k)")
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Tags: Abel, algebra, computation, integration, lemma, mathematical, parts, product, sequence, sum, summation, transformation
Posted in Algebra | No Comments »
Thursday, September 24th, 2009
An Overview of Uniform Convergence
Description
A detailed tutorial of uniform convergence. Step by step tutorial including several example problems of uniform convergence for reference.
Overview
Uniform convergence is a very strong type of convergence, even stronger than pointwise convergence. A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. This concept is important because several properties of these functions are transferred to the limit f if the convergence is uniform.
Tags: converge, convergence, differential equations, functions, limit, Math, pointwise convergence, sequence, speed, uniform convergence
Posted in Differential Equations | No Comments »
Thursday, September 24th, 2009
Overview of the Fibonacci Numbers
Description
A detailed tutorial on the overview of the Fibonacci numbers. Step by step tutorial including several visual representations of the Fibonacci numbers for reference.
Overview
The Fibonacci numbers, also known as the Fibonacci sequence, is one of the most amazing sets of numbers in the world. The set begins with the two numbers 0 and 1 traditionally, although it is also commonly seen starting with 1 and 1 or 1 and 2, and could potentially start anywhere. The Fibonacci numbers follow this pattern:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
This sequence or pattern is that the next number is always the sum of the two number before it. That’s why it start with two numbers…you add them to get the next number, and add that number to the one before it to get the next number, etc. Sometimes this is referred to as a recurrence relation. To mathematically express the pattern, you can write it as:
There is also something called a Fibonacci spiral, which is in the shape of a snail shell. The Fibonacci spiral was found by tracing curves to the corners of the Fibonacci tiles, which have squares whose lengths are successive Fibonacci numbers.
Tags: algebra, Fibonacci, Fibonacci numbers, Fibonacci spiral, Fibonacci tiles, filius Bonaccio, Leonardo of Pisa, Liber Abaci, Math, recurrence relation, sequence, sum
Posted in Algebra | No Comments »
Thursday, September 24th, 2009
Algorithms Explained
Description
A detailed tutorial on the solving of algorithms. Step by step tutorial including several example problems of how to solve algorithms for reference.
Overview
An algorithm is something that you will find at almost any level of math – the more advanced the level of math, the more advanced the algorithm will be. When you use an algorithm, what you are doing is solving a problem by using a finite sequence of instructions. The visual representation of an algorithm is a flow chart…every time you use a flow chart or sequence to solve a problem, even one that isn’t mathematical, you are using an algorithm.
Tags: algorithm, algorithms, calculation, Calculus, data processing, finite, flow charts, instructions, Math, representation, sequence, visual
Posted in Calculus | No Comments »
