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

...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...

...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.
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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...

...BMW: THE 7-SERIES
PROJECT
mary lou
Brief History of Bayerische Motoren Werke:
BMW, a German company is a producer of automobiles and motorcycles. Designed as an aircraft manufacturer and originally founded in 1913 by Karl Fredrich Rapp, the company was commissioned to build the V-12 engine for Austria-Hungary. In need of extra financing, Rapp reconstructed the company as the Bayerische Motoren Werke. In 1917, Rapp left the company and it was taken over by Austrian Franz Josef Popp who later named it BMW. BMW continued their endeavors in producing for the German panzer (airplane) and the motorized division of the German airforce.
It was not until after the World War that BMW was banned from manufacturing for three years. In 1952, they began manufacturing automobiles. In 1959 BMW found themselves in a tough financial situation, as they almost had to sell their interests to Daimler-Benz, the number one auto producer as well as their number one competitor. However, with the opposition of the workforce and labor unions as well as an individual named Herbert Quandt, shares were increased by 50% and the company did a turn-a-round.
As recently as 2004, revenues for BMW were $44 billion dollars with an employee base of 105,972 (HBR 2004). Today, BMW accounts for 10% of the market share with 70% of their sales from Europe and 30% of sales from United States and Japan. These figures equate to the success of BMW and have helped them gain brand...

...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...

...Journey of Life to Heaven
“Uphill” by Christina Rossetti is about the journey of life, or death, to heaven. The poem is an exchange of a series of brief and succinct questions and answers between two speakers: an inquiring traveler who asks many questions about the journey of life or death in which she is heading to (heaven), and an ex-traveler or guide who has taken that path before answering with a assured, and perfectly calm tone. In the poem, the poet uses difference devices such as quatrain, common meter , and perhaps it is written in strict iambic meter with lines vary in length and in the number of feet. The poet uses imagery, and symbolism (allegory)to express emotion and picture a traveler who has to take the road “uphill”, and who hopes to find an inn at the end of her travel. The poem sends a message that though find life hard but there are always comfort, help, and generosity along the way.
This poem illustrates two ways in which line length is varied in a strict meter. The first variation is seen in the first line, which has nine syllables. This is still a five foot line because feet are constructed by iterative parenthesis insertion from right to left, the leftmost foot is the last constructed and can fall short.The second variation involves the number of feet in each line. As can be seen in “uphill” all odd-numbered lines are pentameters, whereas the even-numbered lines vary in length between three and five feet, there are 5 trimeter...

...Come to Freemasons' Hall in Covent Garden, London for the inaugural concert of the Metropolitan Masonic Subscription Concert Series!
London, UK, August 01, 2014 /PressReleasePing/ - Come to Freemasons' Hall in Covent Garden, London for the inaugural concert of the Metropolitan Masonic Subscription Concert Series!
Pianists Samson Tsoy & Pavel Kolesnikov will play an exciting programme for two pianos including the well-known Saint-Saens' "Carnival Of Animals", together with other popular compositions by Ravel, Tchaikowsky and Rachmaninov.
The Metropolitan Grand Master will be present, making this occasion the more exceptional. The concert is open to the general public, who will have the rare opportunity of attending a concert in Freemasons' Hall.
Come to Freemasons' Hall in Covent Garden, London for the inaugural concert of the Metropolitan Masonic Subscription Concert Series on Tuesday October 7th!
Proceeds will go to charity.
Complimentary wine and canapes will be served at the interval.
The Bookitbee story
Kenton and Frank, Bookitbee founders, have been friends since school. Although very different characters, what they both had in common was an incredible entrepreneurial drive and friendship that led them to where they are today. Bookitbee is now fast becoming the number one choice for UK based event organisers looking for an easy and competitively priced booking and ticketing system.
The Foresight Group...

...History of the World Series - 1919
Unfortunately, Francis C. Richter was a little off the mark. The 1919 World Series was, in fact, not honorably played by every participant, as was disclosed late in the 1920 season when confessions were made.
Eight members of the 1919 White Sox -- pitchers Eddie Cicotte and Claude (Lefty) Williams, outfielders Joe Jackson and Happy Felsch, first baseman Chick Gandil, shortstop Swede Risberg, third baseman Buck Weaver and reserve infielder Fred McMullin -- were charged with conspiring to fix the outcome of the fall classic against the Cincinnati Reds. The eight became forever known as the "Black Sox."
A sharp shift in the betting odds shortly before the start of the World Series -- the highly favored White Sox suddenly became underdogs -- aroused curiosity, as did swirling rumors that something might be amiss in certain players' onfield effort. But, overall, fans and other observers accepted the "public presentation" of the 1919 Series. Perhaps, as apparently was the case with Richter, they saw only what they wanted to see.
What everyone saw in Game 1 was a scintillating performance by the Reds' Dutch Ruether. Whatever "assists" he might have received from various members of the opposition, Ruether pitched a complete-game six-hitter and went 3-for-3 at the plate (two triples) with three runs batted in. Outfielder Greasy Neale, who went on to lead the Reds in hitting in...

...Experiment on Metals Aim-To find out which of these metals will be most reactive with hydrochloric acid,and plan an investigation to test it out.The Metals given:Calcium,Aluminium,Iron,Magnesium and Zinc. Prediction-I predict that the metal will be the most reactive with hydrocholoric acid is calcium because calcium is in Group 2 which is the alkaline earth metals in the periodic table and because calcium is at the extreme left side of the periodic table and as you go across a period,the element change from reactive metals on the left to non reactive metals on the right.On the extreme right of the periodic table is the noble gases.Also calcium is Group 2 which is the alkaline earth metals and they are part of the reactive metals which is Group 1 and 2.Also out of all of the metals given calcium reacts more faster with air and creates a sparkling sight and can damage your eyes creating calcium oxide in word equation it is Calcium+Oxygen equals calcium oxide.In chemical equation it is 4Ca +O2 equals 2 Ca2o.Calcium also reacts more than zinc,aluminum,iron and magnesium with water and much more violent and quicker creating vapour and moves much faster than the other metal and creates a vigrous sizzling sound and fizzing in water and dissloves much faster than the other metals. The alkaline metals get more reactive as you go down the group and calcium is further down than magnesium so therefore it must be more reactive than magnesium. Calcium is in group 2 in the periodic table...