The Balmer series is characterized by the electron transitioning from n ¡Ý 3 to n = 2, where n refers to the radial quantum number or principal quantum number of the electron. The transitions are named sequentially by Greek letter: n = 3 to n = 2 is called H-¦Á, 4 to 2 is H-¦Â, 5 to 2 is H-¦Ã, and 6 to 2 is H-¦Ä. As the spectral lines associated with this series are located in the visible part of the electromagnetic spectrum, these lines are historically referred to as H-alpha, H-beta, H-gamma and H-delta where H is the element hydrogen.

Balmer Series (Second) (visible light) n=2 limit = 365 nm

n = 3, ¦Ë = 656.3 nm, ¦Á, color emitted: red
n = 4, ¦Ë = 486.1 nm, ¦Â, color emitted: bluegreen
n = 5, ¦Ë = 434.1 nm, ¦Ã, color emitted: violet
n = 6, ¦Ë = 410.2 nm, ¦Ä, color emitted: violet
Although physicists were aware of atomic emissions before 1885, they lacked a tool to accurately predict where the spectral lines should appear. The Balmer equation predicts the four visible absorption/emission lines of hydrogen with high accuracy. Balmer's equation led physicists to find the Lyman, Paschen, and Brackett series which predicted other absorption/emission lines found outside the visible spectrum.

The familiar red H-alpha line of hydrogen which is the transition from the shell n=3 to the Balmer series shell n=2 is one of the conspicuous colors of the universe contributing a bright red line to the spectra of star forming regions.

Later, it was discovered that when the spectral lines of the hydrogen spectrum are examined at very high resolution, they are found to be closely-spaced doublets. This splitting is called fine structure. It was also found that excited atoms could jump to the Balmer series n=2 from orbitals where n was greater than 6 emitting shades of violet.

[edit] Balmer's formula
Balmer noticed that a single number had a relation to every line in the hydrogen spectrum that was in the visible light region. That number was 364.56 nm. When...

...Fourier series
From Wikipedia, the free encyclopedia
Fourier transforms
Continuous Fourier transform
Fourier series
Discrete-time Fourier transform
Discrete Fourier transform
Fourier analysis
Related transforms
The first four partial sums of the Fourier series for a square wave
In mathematics, a Fourier series (English pronunciation: /ˈfɔərieɪ/) decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.
Contents
[hide]
Definition[edit]
In this section, s(x) denotes a function of the real variable x, and s is integrable on an interval [x0, x0 + P], for real numbers x0 and P. We will attempt to represent s in that interval as an infinite sum, or series, of harmonically related sinusoidal functions. Outside the interval, the series is periodic with period P. It follows that if s also has that property, the approximation is valid on the entire real line. The case P = 2π is prominently featured in the literature, presumably because it affords a minor simplification, but at the expense of generality.
For integers N > 0, the following summation is a periodic function with period P:
Using the identities:
Function s(x) (in red) is a sum of six sine functions of different...

...hp calculators
HP 50g Using Taylor Series
The menu LIMITS AND SERIES Practice using Taylor series
hp calculators HP 50g Using Taylor Series The menu LIMITS AND SERIES The menu LIMITS AND SERIES contains commands related to limits. To access it you press !Ö. You are presented then the calculus menu as a CHOOSE box:
Figure 1
Its second menu item is 2.LIMITS AND SERIES... You can use such CHOOSE boxes much like menus of computer applications. You can move the selection using the arrow keys. You can also jump to a certain menu item by typing the first few letters of the command or the number at the left of the menu item. Pressing the `key or the menu key %OK% will execute the selected item. In this CHOOSE box you press 2 to select 2.LIMITS AND SERIES.. and then ` or %OK% to display the menu:
Figure 2
The command DIVPC needs two polynomials and an integer. It returns the increasing power quotient of the two polynomials up to an order indicated by the integer. The command lim takes an algebraic object and an equation of the form variable=expression. It returns the limit of the algebraic expression when the given variable approaches the expression at the right hand side of the equation. The command SERIES needs an algebraic expression, and equation of the form variable=expression, and an integer. It returns a list at stack level 2 and...

...
5a.
When y = sin (1), y = 0.841. Using the Taylor series with two terms, y = 0.830.
When y = sin (5), y = -0.958. Using the Taylor series with two terms, y = - 15.8.
When y = cos (1), y = 0.540. Using the Taylor series with two terms, y= 0.500.
When y = cos (5), y = 0.284. Using the Taylor series with two terms, y = - 11.5.
By using the formula, Percentage Error =
Percentage Error for Taylor series with two terms =
= 1377.18 % ≈ 1380 % (3sf)
5b.
When y = sin (1), y = 0.841. Using the Taylor series with three terms, y = 0.842.
When y = sin (5), y = -0.958. Using the Taylor series with three terms, y = 10.2.
When y = cos (1), y = 0.540. Using the Taylor series with three terms, y = 0.542.
When y = cos (5), y = 0.284. Using the Taylor series with three terms, y = 14.5.
Percentage Error for Taylor series with two terms=
= 1492.80% ≈ 1490% (3sf)
5c.
When y= sin (1), y = 0.841. Using the Taylor series with four terms, y= 0.841.
When y = sin (5), y = -0.958. Using the Taylor series with four terms, y= -5.29.
When y = cos (1), y = 0.540. Using the Taylor series with four terms, y = 0.540.
When y = cos (5), y = 0.284. Using the Taylor series with four terms, y = - 7.15.
Percentage Error for Taylor series with...

...MATH 152 MIDTERM I 02.11.2012 P1 P2 P3 Name&Surname: Student ID: TOTAL
Instructions. Show all your work. Cell phones are strictly forbidden. Exam Duration : 70 min. 1. Show that 1 p n (ln n) n=2 converges if and only if p > 1: Solution: Apply integral test: Z Z
ln R 1 X
R
2
1 p dx x (ln x) p=1 p 6= 1
let ln (x) = u then
ln 2
so that when p = 1 and p < 1 integral diverges by letting R ! 1, so does the series. When p > 1 then integral converges to ! 1 p 1 p 1 p (ln R) (ln 2) (ln 2) lim = , R!1 1 p 1 p 1 p so does the series. 2. (18 pts.) Find the in…nite sum 1 : n (n + 2) n=1 Solution: See that 1 1 = n (n + 2) n 1 n+2
1 X
8 R < ln ujln 2 ln 1 ln R du = 1 p : u p up 1
ln 2
hence 1 n (n + 2) n=1
k X
= =
n=1
1 1 1 1 1 + + + :::: 3 2 4 3 5 1 1 1 1 1 + + + k 2 k k 1 k+1 k 1 1 1 = 1+ + 2 k+1 k+2 1 ! 3 2
k X
1 n
1 n+2
1 k+2
so that
k X 1 1 = lim n (n + 2) k!1 n=1 n (n + 2) n=1
1 X
= lim
k!1
1 1 + k+1 k+2
=
3 2
1
3. (18 pts.) Find the Taylor series for f (x) = ln x at x = 4. Determine its interval of convergence. Solution: Recall that
1 X n
tn
= = =
1 1 t
;
n=0 1 X
jtj < 1 jtj < 1 jtj < 1
( 1) tn
n=0 1 X ( 1)n tn+1 n+1 n=0
1 ; 1+t
ln (1 + t) ;
let x
4 = t then ln (x) = = ln (4 + t) = ln 4 + ln 1 + ln 4 + t 4 = ln 4 + jx
1 X ( 1)n t n+1 4 ; n+1 n=0
t...

...Each individual has his or her own ideas about many things, and there is no exception on the perception of a better life. Perception of a better life is not just about having a luxurious house or expensive car. Someone can be categorized as a successful person if he or she does not just have many expensive things, but also has good social status in society.
To stand out and become well known in the public is certainly not an easy thing to achieve. Someone must have the courage to sacrifice what he or she has in order to get ‘recognition’ in the society. The desire to stand out and become well known can be found in many countries, including an advanced country like America. Americans realize that they live in super power country. Therefore, they begin to raise their prestige in their daily lives. Maintaining the image as a country that has a high prestige is more difficult than to get the image. One way for people who live in America to maintain that image is to look ‘different’ and attractive.
Keeping fit-looking appearance that reflects someone’s personality is essential to maintain the prestige. An expensive and attractive appearance is very important because it is an outside representation for someone, to show to other people that he or she is a person who has a good wealth, even though perhaps, it is not. When appearance becomes a main thing for someone to declare their existence, many people are competing to look attractive and different from others. Expensive...

...Baseball is said to be America’s favorite pastime, and for me that is true. The definition of baseball is a game played with a bat and ball by two teams of nine players each, the object being to score runs by advancing runners around four bases. (The McGraw-Hill Children’s Dictionary). Baseball is usually played in the summer. St. Louis Cardinals is my favorite team. Last time they won the World Series was in 2011. Which was a happy time for me because that meant they were the best? They are pretty good this year and they could possibly win again. They have a good record, and they have players that love the sport as much as me.
Baseball is unlike any other sport. In baseball you have to have skills and patience. You just can’t join if you don’t know what you are doing so you have to know what you doing to able to play the game. Baseball has coaches and umpires to help you with the game. The coach teaches you the rules of the game, and how to play the game. Umpires are the people that enforce the rules during the game.
The rules of a baseball game are as followed. There are three strikes out and four balls for a person. If you get three strikes that means you are out. In the game players on the batting team take turns hitting against each other. But if you get four balls that means you get to go to first base. If you catch the ball when it is hit that means the batter is out. When you score you get one point. There are nine innings during the game. There...

...a market. These methods are most appropriate when there is not much historical data to work with.
2. Causal methods assume that demand is strongly related to a particular cause, such as environmental or market factors.
3. Time series methods are based on the assumption that historical patterns of demand are a good indicator of future demand, and that over a period of time, demand can be charter in three different ways: as an underlying trend (flat, up , or down), as a circle (daily, weekly, seasonally , and so on), and as irregular fluctuations (peaks or valleys) over time.
4. Simulation methods are a combination of causal and time series methods will imitate the behavior of consumers under different circumstances.
With the product of Cadbury’s Roses boxed chocolate, Purchasers of Costco’s confectionery Dept will forecast the consumption of this product base on 4th methods – simulation – that will be described in detail as below:
Simulation method comprises 2 methods causal and time series.
Base on causal method, chocolate’s demand is effected too much by:
1. Events: Christmas, Mother Day,
2. Price: better than other distributors or retailers.
3. Reputation of products and distributors.
In the other hand, with time series method, chocolates consumption forecast is subjected to:
* 4. Last year sales history.
* 5. Market survey of chocolate consumption in Australia.
4.9.3...

...[pic] Fourier Series: Basic Results
[pic]
Recall that the mathematical expression
[pic]
is called a Fourier series.
Since this expression deals with convergence, we start by defining a similar expression when the sum is finite.
Definition. A Fourier polynomial is an expression of the form
[pic]
which may rewritten as
[pic]
The constants a0, ai and bi, [pic], are called the coefficients of Fn(x).
The Fourier polynomials are [pic]-periodic functions. Using the trigonometric identities
[pic]
we can easily prove the integral formulas
(1)
for [pic], we have
[pic]
(2)
for m et n, we have
[pic]
(3)
for [pic], we have
[pic]
(4)
for [pic], we have
[pic]
Using the above formulas, we can easily deduce the following result:
Theorem. Let
[pic]
We have
[pic]
This theorem helps associate a Fourier series to any [pic]-periodic function.
Definition. Let f(x) be a [pic]-periodic function which is integrable on [pic]. Set
[pic]
The trigonometric series
[pic]
is called the Fourier series associated to the function f(x). We will use the notation
[pic]
Example. Find the Fourier series of the function
[pic]
Answer. Since f(x) is odd, then an = 0, for [pic]. We turn our attention to the coefficients bn. For any [pic], we have
[pic]
We...