Posts Tagged ‘summation’
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 »
Tuesday, October 6th, 2009
How to Test for Convergence Using the Geometric Series Test
Description
A detailed tutorial on how to test for convergence using the geometric series test. Step by step tutorial including several examples of testing for convergence using the geometric series test for reference.
Overview
A geometric series is a series that maintains a constant ratio between a set of terms. This series is an addition series, and would be expressed as 1/a + 1/2a + 1/4a, extending as far as you wish in either direction. If a series does not have that constant ratio, then it is not a geometric series. The series should converge at one, because as all the numbers are added they get closer and closer to one. The first term of a geometric series is given by a, and the ratio of a geometric series is given by r. If the ratio is less than one, then the geometric series converges to a / (1 – r). If the ratio is greater than or equal to one, then the series diverges. Usually the series will converge, which is why this is considered a test for convergence and not for divergence.
Tags: a, addition, Calculus, converge, convergence, diverge, divergence, equal to, first term, geometric, greater than, less than, Math, notation, r, ratio, series, summation, test
Posted in Calculus | No Comments »
Thursday, October 1st, 2009
Introduction to Summation
Description
A detailed tutorial of summation. Step by step tutorial including several examples of summation for reference.
Overview
Summation is a simple concept – notice its similarity to the word sum. In simple terms, summation simply means the addition of a set of numbers, the result being their sum or total. The summation notation is especially useful when you are given a large set of numbers to work with. The summation notation is expressed as a large capital Sigma (a Greek letter), and when given number values the notation would look like this:
Tags: addition, algebra, associative, commutative, divergent, extrapolation, interim, Math, series, sigma, sum, summation, total
Posted in Algebra | No Comments »
