xerxes02's New Writeups

Alternating series (thing)

2002-12-12T21:00:07Z

Definition

An alternating series is a series containing terms with alternating signs. Here are two examples:


 ∞
---     n-1
\   (-1)          1   1   1
/   ------- = 1 - - + - - - + ...
---    n          2   3   4
n=1

 ∞
---     n
\   (-1) n     1   2   3   4
/   ------ = - - + - - - + - + ...
---  n + 1     2   3   4   5
n=1

Looking at our examples we notice that the n-th term of an alternating series can be described in two ways:

a_n=(-1)^n-1b_n or a_n=(-1)ⁿb_n

Where b_n is a positive number. More generally, b_n = |a_n|.

Alternating Series Test

As is with all series mathematics, the fundamental question we must ask is, "Does this series converge?". With…

voltaic cell (thing)

2002-12-10T03:26:37Z

Definition

What is a voltaic cell? A voltaic cell is a device in which chemical energy is converted into electric energy. This is accomplished by submerging two dissimilar metals in an electrolyte, connecting the two peices of metal with a wire or other conductive substance, and a salt bridge. The voltaic cell is also reffered to as a galvanic cell, paying homage to the two 18th century scientists who pionerred the reasearch into this topic, Alessandro Volta and Luigi Galvani.

Introduction

Voltaic cells also rely on an important type of reaction: oxidation-reduction reactions, also called redox reactions. In an oxidation-reduction reaction, two substances interact by transferring electrons. Oxidation is defined by a loss of electrons, Reduction is a gain of electrons. This can be remembered by the mnemonic "LEO the lion says GER". Loss of electrons - O…

Platonic metaphysics and the poetry of Emily Dickinson (idea)

2002-12-08T06:45:48Z

Examining the fields of Philosophy and English, one can say without a doubt that two extremely influential people in these two areas were Plato for his work in Greek philosophy, formulating the "Ideal Form" and recording Socrates' dialogues, and Emily Dickinson for her contributions to American Letters, specifically through her poetry. But one would rarely think of these two well known figures in relation to each other, if not only for the vast ocean of time that separates them – over two millennia. Examining issues common to both poets, one may find more in common than at first glance; the issue of phenomenology or interpretation, concepts of imagination and knowledge as they relate to inspiration and poesy, and finally a formal treatment of tension due to time-space and death tropes as an extension of the metaphysical Ideal. Through a synthesis of Plato's groundbreaking ideas on universal archetypes and recurring themes such as death and impermanence in…

Intimations of Immortality (thing)

2002-04-13T16:52:21Z

Full Title: "Ode: Intimations Of Immortality From Recollections Of Early Childhood"

Information:

Author: William Wordsworth

Original Text: Poems in Two Volumes

Published: 1807

Genre: Romantic

Full Text:

                  The child is father of the man;
                  And I could wish my days to be
                  Bound each to each by natural piety.
                       --(Wordsworth, "My Heart Leaps Up")                  


                                   I

          THERE was a time when meadow, grove, and stream,
          The earth, and every common sight,
                    To me did seem
                  Apparelled in celestial light,
          The glory and the freshness of a dream.
          It is not now as it hath been of yore;--
                  Turn wheresoe'er I

…

Implicit Differentiation (thing)

2002-04-10T02:05:38Z

Implicit Differentiation

Definition: A process for finding dy/dx when y is implicitly defined as a function of x by an equation of the form ƒ(x,y) = 0.

What Does this Mean?

In other words, Implicit Differentiation is a method to find the derivative of a function when separation of x and y is not possible, or one is unable to put a function in the familiar "y=" or "ƒ(x)=" forms. Take, for example, the relation y = x² + xy³. Now say one needs to find dy/dx. By basic differential techniques, this is not possible. This is where Implicit Differentiation comes in.

That's Great, but How Does One Accomplish This?

Relying on our foreknowledge of algebra, calculus, and basic diffentiation (duh!), one looks at y as a…

Taylor Series (thing)

2002-04-09T03:41:34Z

Taylor Series

Definition:
A Taylor Series is a polynomial function with an infinite number of terms, expressed as an Infinite Series. Taylor Series can be used to represent any function, as long as it is an analytic function. If the function is not infinitely differentiable, Taylor Series can be used to approximate values of a function. Either way, the approximation will be more accurate along a certain interval of convergence.

Taylor Series Basics
To understand Taylor Series, let's first construct a polynomial: P(x) = a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴ + ... + a_nxⁿ + ... or, in other words

 ∞
---    n
\   a x
/

…