Harmonic Series Investigation
This investigation will examine some aspects of Harmonic Series’. The name harmonic series come from overtones or harmonics in music. This is because the wavelengths in the overtones on a vibrating string are ½, ⅓. ¼ etc. just like the terms in the harmonic series. However, this investigation will mainly focus on testing for convergence or divergence on the harmonic series, as well as other variations of the series rather than the physics behind the numbers. The numbers in the harmonic series have been used for many architectural designs especially during the Baroque period. Harmonic series’ have some interesting properties which I shall i explore throughout this paper.
The Harmonic series can be represented by n=11n= 1 + 12+ 13...1
First I will prove the the Harmonic Series is divergent. The harmonic series seems as if it would converge because each of its terms approach zero, but it is actually divergent. I will use the first few terms of the harmonic series and compare it to the first few terms of another divergent series. The series I compare the harmonic series to was created because it diverges and the sum of each fraction with the same denominator is equal to ½. The divergent series will be used as a known divergent series to be compared to in each of the divergence tests throughout this paper.
Harmonic Series - 1+12+13+14+15+16+17 +18...
Divergent Series - 1+12+14+14+18+18+18+18…
The second series can be proven to diverge by grouping the terms that contain the same denominator ie. ¼ and ¼. The sum of the terms in each of these groups is equal to ½. This combination makes the series appear as this : 1+(12)+(12)+(12)+(12)…
The series above is divergent because the halfs add up infinitely ther...
... middle of paper ...
...uared ie.x=11n2= 1+14+19+116…
This series is convergent using the p-series test because the value of p=2 and when p>1, the series is convergent.
In this portfolio, I have investigated information about the harmonic series and then some variations of harmonic series’. To summarize, I have concluded that the harmonic series in divergent through comparing it to another known divergent series and through the improper integral test. I have proved divergence for a few other invented series through the same tests. I also showed the alternating harmonic series and proved that it converged to ln2 using the taylor series. I then proved that any series in the form n=11an+b is divergent and its sum is infinite. Finally in this investigation i modified the harmonic series by putting an exponent in the denominator and using the p-series test to prove convergence or divergence.
* Question 2. Given the sequence S = {-9, 2, 4, 6, 30, -10, 1, 5, 8, 7},
S/F/36. IPO valuation and analysis This work presents classical analysis of the Initial Public Offering (IPO). First of all, the general financial position of the company and the quality of management are scrutinized. This is an important step in the analysis as it allows approaching the valuation step with all necessary adjustments made beforehand. Then the valuation process itself is conducted. The author uses post-IPO cash-flow analysis in order to allow for substantial reduction of debt due to the IPO. CAPM and WACC concepts are utilized to obtain the value of the company. However, this work is not only useful for IPO valuation. The author makes comprehensive analysis of benefits and disadvantages of the IPO. The role of the underwriter and qualities it has to possess are also discussed. Since there may exist the phenomenon of short-run overperformance and long-run underperformance, the analysis of stock market returns is accomplished. Finally, the appropriateness of different stock exchanges for different types of company is discussed. The paper will be useful for students doing comprehensive case-study of the IPO.
This paper will be explaining the history behind the creation of the harpsichord. This paper will also look at some of the famous harpsichord composers throughout history. There are many other interesting instruments in the world, but the harpsichord made it possible to create many different sounds that led to even greater compositions. The harpsichord was the quintessential instrument in producing the modern days string instruments that are seen in the world today. The harpsichord made it possible to create legendary musical pieces. The harpsichord was one of the main instruments used during the time known As the Baroque Period.
Throughout this research paper the topic is going to be along the lines of the Baroque Art in Europe and North America, which comes from chapter nineteen of our Art History book. The main purpose is to review major ideas and principles in this chapter by writing an analysis of certain points that were highlighted. For example, certain techniques that were used to define the Baroque Art, major sculptures, architectures, and paintings, and also just some general background information about this time period. I decided to write on this subject because when reading the chapters, the Baroque period seemed to catch my attention the most due to the amazing architecture that was built during this time.
We can see that in 1990 dividend payout ratio was increased sharply compare to the previous years. Also, we can see that FPL had a loss in 1990, but the company still increased dividend. Furthermore, in 1991 to 1993 dividend payout ratio was significantly high when compare to the historical data. These sharp changes ...
When watching the movie “Finding Your Roots” you get a deeper understanding of what life was like during slavery years and how people lived their lives. But not only do you get a deeper feel for what it was like for the people, you also think about how your own family lives were at the time on slavery. We think we have an idea of what the slavery days was like for our ancestors and have a general thought process about timeline of events. Many of our ancestors have made the way for us to live our lives with freedom and to be treated as human. We know our families were once sold and used as property and not people. But do we actually know about the era that profound our lives today and the roots we come from?
The health of the Singapore economy based on the analysis of its positive balance of payment is very favourable because it is shown in part (a) that Singapore economy is growing at a high rate and is assured of economic growth. Singapore has very few debts and these debts will not hinder further plans as the income generated will be more focused on research and development sector rather than repayment of debts. Additional, Singapore’s Gross Domestic Product (GDP) is high and this means that the production activities are also high and the people within
The article Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy was written in 1968 by Edward I. Altman. The purpose of the article is to address the quality of ratio analysis as an analytical technique. At the time some academicians were moving away from ratio analysis and moving toward statistical analysis. The article attempted to determine if ratio analysis should be continued, eliminated and replaced by statistical analysis or serve together with statistical analysis as cofactors in financial analysis. The example case used by the article was the prediction of corporate bankruptcy.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
The time value of money serves as the foundation for all other notions in finance. It affects business finance, consumer finance and government finance. Time value of money results from the concept of interest. The idea is that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received. Time value of money can be illustrated by the fact that a dollar received today is worth more than a dollar received a year from now because today's dollar can be invested and earn interest as the year elapses. Implicit in any consideration of time value of money are the rate of interest and the period of compounding. This paper will list various financial applications of the time value of money and explain the components of the discount/interest rate.
Fibonacci sequences are set of numbers based on the rule that each number is equal to the sum of the preceding two numbers; it can be also evaluated by the general formula where F(n) represents the n-th Fibonacci number (n is called an index), the sum of values in pascal`s triangle diagonal also demonstrates Fibonacci sequences. The presentation and report are designed to discover the application of Fibonacci sequences in daily life. The famous Fibonacci sequence has captivated mathematicians, artists, designers, and scientists for centuries therefore it is suggested as an important fundamental characteristic in real life.
A mathematician and scholar by the name Frenchman Nicole Oresme used the system of rectangular coordinates as well as perhaps the first time-speed-distance graph. He also was the first to use fractional exponents, and also worked on infinite series, being the first to prove that the harmonic series (1⁄1 + 1⁄2 + 1⁄3 + 1⁄4 + 1⁄5.) is a divergent infinite series. Harmonic series helps to understand wavelengths in music, which also relates to physics.
The modern music notation was made during this period. Catholic monks during the Baroque Period developed the starting forms of music notation to standardize sacred musical compositions in their church. Pitches of notes are indicated with how notes are placed in between lines or spaces. The duration of each note is determined by the shading of the circle, the stems and the tails on notes (Paterson, 2017). Because of this unified “language”, reading and composing music is made easier for musicians.
The observations of financial data of a number of companies is meticulously studied and their similar characteristics are found by one 's mind. If the data have repetitive relationships, then some principles can be formed and in some cases new thoughts can be induced. If the observations are not influenced by current principles and actions, more innovative thoughts can be induced. For example, based on historical data, the ratio of received cash to sales in several companies related to the same industry might suggest a certain trend. Based on this ratio, we can predict the future cash revenues. This idea is falsifiable. In other words, the macro economic factors such as inflation rate, the volume of the liquidity, the interest rate of bonds, and GDP might affect the trend of the cash to sales ratio in industry and the future revenue collection might not be predictable based on the past trend of sales (Saghafi,