...SEQUENCES AND SERIES Finite Series: expression of the form [pic]where each term is formed by some definite rule. Infinite Series: it is of the form [pic] (Example: infinite geometric series with a0=1 and r=1/2) [pic] Sum of Infinite Series (Sn) ❖ The sum of finite series is merely the algebraic sum of all term. (addition) ❖ Infinite series has no sum in ordinary sense. Definitions: The sum of an infinite series is the limit, if it exist, of the sum of a finite number of terms, as the number of terms approaches infinity. ([pic] (Sum of infinite geometric progression) Convergence and Divergence If the series has a sum Sn i.e. if Sn approaches a limit when [pic] Power Series An infinite series of the form [pic] (1) in which a0 , a1, a2, . . . , an, . . . are constants and x is a variable, is called a power series in x. The totality of all values of x for which a power series converges is called its interval of convergence. This interval of convergence always includes the value x = 0, and its range is determined by the ratio test, R = [pic]. Examples. Find the interval of convergence for each of the following series. 1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic] Maclaurin’s Series A power series of the form [pic]. is called Maclaurin’s series, or the power-series expansion of f(x) about x = 0. It is named for the Scottish...
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...on U.S. roads (American Petroleum Institute). With the ever-increasing presence of natural gas in our lives, it is important to know how it affects us on the consumer level. More importantly, it is essential to understand how it affects us monetarily. When we can understand trends and forecast the pricing of natural gas, we can improve our financial planning, resource allocation, etc. II. Data Set Description and Methods Used This paper will be conducting a time series analysis on U.S. Natural Gas prices from January 2002 through December 2012. Prices were collected by the Energy Information Administration on a monthly basis, and the prices are measured in dollars per thousand cubic feet. The raw data set is much more extensive, and measures data such as wellhead price, import and export price, etc. I chose to neglect this in my data set for simplicity sake and because it is more applicable to look at prices directly affecting everyday individual consumers. The methods used in the time series analysis of U.S. Natural Gas Prices include non-seasonal differencing, exploring autocorrelation and partial autocorrelation functions, ARIMA modeling, conducting diagnostics on modeling by looking at residual normality [QQ plot, Shapiro Test], Box-Pierce...
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...Time series analysis We are pleased to submit the following report on the “Time Series Analysis”. By completing the report, we have got acquainted importance and relevance of time series on business application. We also perceived idea on the whole process of Time Series Analysis. We acquired knowledge about the method of measuring trend, growth rate, acceleration rate etc. In spite of limitation of time & opportunity we have tried our level best to complete the report. We are pleased to provide you with this report with necessary analysis, references and we shall be available for any clarification, if required. Thank you for assigning us in this study. On behalf of the group Md. Arif Hasan ID: 12-150 Table of Contents Serial No Topic Page No 1 Letter of Transmittal 1 2 Rationale of the study 2 3 Objectives of the report 3 4 Methodology of the report 3 5 What is Time Series 4 6 Uses of Time Series in Business 5 7 Components of a Time Series 5 8 Classical Time Series Model 9 9 Methods of trend measurement 9 10 Least squares method 10 11 The Growth Rate 14 12 The Acceleration Rate 15 13 Rule of 72 16 14 Bibliography 17 Rationale of the study Having been assigned to prepare a report on Time Series Analysis we are submitting the term paper based on our findings and understandings. Time series analysis has vast application and is of huge importance in the field of Business and Economics as well as in decision making thereof. Calculating secular trend we can...
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...Time Series and Forecasting Learning Team A Quantitative Reasoning for Business/501 August 23, 2011 Dr. Champion Time Series and Forecasting The purpose of this paper is for statisticians from the accountant department of Norton Company to compute the quarterly seasonal index for the years of 2003 through 2006 by using the ratio-to-moving-average method. In addition, the accountants will deseasonalize the data and determine the trend equation. Furthermore, the statisticians will estimate Norton Company’s seasonally adjusted sale for the four quarters of 2007. The quarterly sales for the Norton Company were given in millions of dollars for four years. Therefore, the recorded quarterly sales for the Norton Company were referenced and included 16 quarters total. A scatter plot was graphed showing the recorded sales for the 16 quarters. The following equation was obtained from the historical data: y = 0.461x + 4.2 with the R² of 0.1695 and a R of 0.412. R is the coefficient of correlation, which provides the strength and direction of a linear relationship. R2 is the coefficient of determination, which measures the amount of variation in y that can be explained by the variation in x for example: 0 ≤ R2 ≤ 1. With this unadjusted regression equation and R² we can see that there is seasonality. Accordingly, a higher R² is required to prove that our equation is accurate and good for forecasting. The first step is to find the moving total....
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...Time Series Analysis Yt=observed value of the time series in time period t TRt=the trend component or factorin time period t SNt=the seasonal componentor factorin time period t CLt=the cyclical componentor factorin time period t IRt=the irregular componentor factorin time period t 7.1) CL*IRCL=IR a) SN1=1.191 TR1=240.5 CL1=null IRt=null SN2=1.521 TR2=260.4 CL2=0.998 IR2=0.990 SN3=0.804 TR3=280.4 CL3=0.994 IR3=0.986 SN4=0.484 TR4=300.3 CL4=1.003 IR4=1.008 b) It presents a multiplicative decomposition model Yt=TRt*SNt*CLt*IRt SNt*IRt=YtTRt CLt snt*irt=YtCMAt =YtCMAt Equation of the estimated trend: TRt=Bo+B1t dt=B0+B1t+εt TRt=220.53+19.94(t) c) Yt=trt*snt Y17=220.53+19.9417*1.191=666.6 Y18=220.53+19.9418*1.521=881.6 Y19=220.53+19.9419*0.804=482.1 Y20=220.53+19.9420*0.484=299.9 d) Yt=trt*snt*cl We cannot see a definitive cycle and because the values of cl are close to 1. We do not take it into account. Y21=220.53+19.9421*0.191=761.6 e) Since there are just four years of data and most values are near 1 we cannot discern a well-defined cycle. f) Y21=220.53+19.9421*0.191=761.6 It agrees with the values computed in part c g) Excel Spreadsheet h) Prediction intervals for the next 4 quarters t=17,18,19,20 t=17:654.094,679.542 t=18:869.038,894.542 t=19:469.107,494.556 t=20:286.977,312.426 8.1) Smoothing equation l0=t=1nYtn Which is the average of the first series values lT=αyT+(1-α)lT-1 α:smoothing constant ...
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... NIT TRICHY AN ASSIGNMENT ON TIME SERIES ANALYSIS PLANNING AND CONTROL OF OPERATONS FAWAZ MOHAMED KUTTY 215112035 MBA Ist YEAR TIME SERIES ANALYSIS A time series may be defined as a set of values of a variable collected and recorded in a chronological order of the time intervals. Time series is used by statisticians to describe the flow of economic activity. In short time series refers to the data depending on time. It refers to a set of observations concerning any activity against different periods of time. The duration of the time period may be hourly, daily, weekly, monthly or yearly. According to Morris Hamburg “A time series is a set of statistical observations arranged in chronological order”. Therefore time series is also called historical data or historical series. The study of movement of quantitative data through time is referred to as ‘time series analysis’. Time series is of great importance to the planners of economic development and economists. The success of planning depends upon making accurate forecasts of future conditions of economy. It enables the economists to foresee what is likely to come and to analyze the repercussions of past behavior. The analysis of time series enables us to understand the past behavior or performance. Time series analysis can be used to know how the data changes over time and find out probable reasons for such change. UTILITY OF TIME SERIES ANALYSIS Analysis of time series is of relevance whenever a variable is found...
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...REPORT ON TIME SERIES ANALYSIS REPORT ON TIME SERIES ANALYSIS SUBMITTED TO M. KHAIRUL HOSSAIN PROFESSOR Department Of Finance University Of Dhaka SUBMITTED BY Group – 17 Section-A BBA 12th Batch Department Of Finance WE ARE... |Sl. No |Name |Roll No | |1. |Dulal Paul |12-143 | |2. |Rahat Hussain Md. Zaidy |12-149 | |3. |MD. Arif Hasan |12-150 | |4. |MD. Khurshid Alam |12-170 | |5. |MD. Saiful Islam |12-254 | Letter of Transmittal Date: 16th September, 2008 M. Khairul Hossain Professor Department Of Finance Faculty of Business Studies University of Dhaka Subject: Submission of report We are pleased to submit the following report on the “Time Series Analysis”. By completing the report, we have got acquainted importance and relevance of time series on business application. We also perceived idea on the whole process of Time Series Analysis. We acquired knowledge about the method...
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...CASE ASSIGNMENT: NFL International Series 1. Discuss the NFL’s current strategy for global market entry. How has this strategy changed and why did the NFL make these changes? The strategy for the NFL’s global market entry was the NFL Europa league which closed down in 2007 with no plans of reopening. The original strategy was to send teams to London to play one game a season to help expand the idea of American football. But obvious changes had to be made due to the fact that the league lost the NFL money for 15 years straight and was only used to dump off ok players during training camp. Instead of the teams arriving to the game a day before; they arrived at the beginning of the week and held open practices that media was granted access too. The biggest boosts was the agreement reached by the league with the BBC to broadcast the game live, which would potentially add an additional 2 million viewers, which before the games could only be viewed on a British pay channel( only about 120,000 viewers). By just having the game overseas, it would allow new people to attend the games in which would allow the NFL slowly become global. 2. Discuss the major environmental challenges that the NFL has encountered and how they have addressed these. A major environment change was the NFL teams would be playing in a different country and in a new stadium they had not played in before. Each team is allowed to have practices before the upcoming games. Besides arriving early for...
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...BMW was one of the major players in the luxury car market (retail price above $20,000) having a market share of 10%. Companies in this market segment competed on both tangible and intangible product characteristics. BMW’s focus was on R&D, both in the engineering and styling of its cars, and had built a reputation for high performance engineering. Its launch quality had always been on par or better than that of its major competitors. However, since the late 1980s, the entry of Japanese competition (Honda’s Acura and Toyota Lexus) had set higher standards of conformance in this industry. 1. What are the causes and consequences of BMW’s quality problems with newly launched products? What should be done to improve launch quality? One of the primary reasons for BMW’s quality issues was its strategy of using different tools and materials in the prototyping process when compared to the production runs.BMW supported such a ‘dual’ development process to provide flexibility to its prototyping craftsmen to make changes to the design quickly. For instance, the prototype shop was able to make new parts within a week of being intimated of a design change. In contrast, the lead time for procuring pre-production tools from suppliers was 6 months. Providing such flexibility though had a flip side to it - something which worked in the prototyping phase may not work in the production phase. This meant that potential problems in the design could be identified very late in the development...
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...Time Series Analysis Summary Tokelo Khalema 2008060978 BSc. Actuarial Science University of the Free State Bloemfontein November 1, 2012 Time Series Analysis A time-series is a stochastic process {Xt : t = 1, . . . , T } with a continous state space and discrete time domain. It arises naturally as an ordered series of values observed over time. Examples include daily closing prices of a stock index recorded over several years, say, the flow rate of the River Nile, road casualties in Great Britain over the years 1969-84, etc. Stationary time-series are particularly easy to analyse. A series is stationary if its mean and variance are constant over time. Special aids are available to help determine whether or not a series is stationary. Particularly notable in this regard are the autocorrelation function (ACF) and the partial autocorrelation function (PACF). These are plots of the sample autocorrelation and partial autocorrelation coefficients at various time lags, respectively. If the ACF decays gradually to zero, then the series is non-stationary. If on the other hand the ACF and PACF decay rapidly to zero, then the series is stationary. A series being non-stationary can be brought about by, among others, a trend, irregular fluctuations, or seasonal variation. Non-constant variance, or as commonly called, heteroscedasticity can be eliminated by using a variance-stabilising transformation. A number of ways exist that eliminate a trend. Two of which are, to subtract a regression line...
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...Series Resistance Starters Construction and Operation In this method, resistors are temporarily placed in series with the motor windings during startup. A voltage drop is created across the series resistors during startup, resulting in a lower starting voltage applied to the motor’s terminals. Once the motor has accelerated to normal operational speeds, the resistors may be “removed” and the motor allowed to operate under normal conditions. At startup, the start contacts are closed, forcing the starting currents to flow through the series resistors. Once the motor has been allowed sufficient time to accelerate, the run contacts are closed, thus bypassing the resistors. A timer may be utilized within the control circuit to allow the motor to transition from “start” to “run” operation automatically. Applications 1. Usually on low voltage (less than 600 v). 2. Where current reduction requirements are low, or where load torque during acceleration is minimal. 3. Not often used with large motors because of the high heat loss in the resistors. 4. May be used for full acceleration or for system voltage recovery. Advantages 1. reduced heat loss 2. greater ease of start 3. (If correctly designed and constructed), this method will cause the motor to accelerate the load to almost full speed with the resistors in circuit before they are bridged out. Disadvantages 1. The resistors must be cooled in between starts because of the heat build up in the resistors during start...
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...BMW: The 7-Series Project Case Team 7: Jonghoon Ahn, Sangdong Kim, Chae Won Lee, Wonil Shin, Jonghwan Lee Question 1: How does one define design quality? Answer: Design quality means the product attributes that enhance the utility of the product for the customers. It is that making products for consumers to use the products better with more conveniences and values. Question 2: What are the causes and consequences of BMW’s quality problems with newly launched products? What should be done to improve “launch” quality? Answer: I think there are several reasons in BMW’s new model development way like below; 1. BMW doesn’t use pre-production tools during prototyping. -> BMW has less opportunity to uncover and solve many kinds of quality problems earlier in development process. 2. BMW let suppliers involve into development after “cubing”. -> Suppliers meet quality problems during their first production and they don’t have enough time before new model introduction. 3. When BMW starts the first commercial production, BMW produces new models in old models’ production lines together to reduce fixed costs. -> It causes greater confusion and makes logistics more complex. All these things result in quality problems with newly launched products. To reduce these kinds of things; 1. Use pre-production and production tools earlier to find and to have more time to prevent quality problems expected in commercial production. 2. Involve reliable suppliers earlier...
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...Forecasting Models: Associative and Time Series Forecasting involves using past data to generate a number, set of numbers, or scenario that corresponds to a future occurrence. It is absolutely essential to short-range and long-range planning. Time Series and Associative models are both quantitative forecast techniques are more objective than qualitative techniques such as the Delphi Technique and market research. Time Series Models Based on the assumption that history will repeat itself, there will be identifiable patterns of behaviour that can be used to predict future behaviour. This model is useful when you have a short time requirement (eg days) to analyse products in their growth stages to predict short-term outcomes. To use this model you look at several historical periods and choose a method that minimises a chosen measure of error. Then use that method to predict the future. To do this you use detailed data by SKU's (Stock Keeping Units) which are readily available. In TSM there may be identifiable underlying behaviours to identify as well as the causes of that behaviour. The data may show causal patterns that appear to repeat themselves – the trick is to determine which are true patterns that can be used for analysis and which are merely random variations. The patterns you look for include: Trends – long term movements in either direction Cycles - wavelike variations lasting more than a year usually tied to economic or political conditions...
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...Growing up playing baseball and watching the World Series games on TV, it’s every young player’s dream to be able to be a part of the electrifying experience. In 2011, at the age of 13, I got my chance to attend my first World Series game. It was Game 6 of the World Series between the Texas Ranger and the St. Louis Cardinals. The day of the game, I came home from school flowing with excitement along with a bit of nervousness as this would be my first ever World Series game. Sitting in my room anxiously waiting for my dad to come home from work so that we could head down to the stadium to enjoy the pregame festivities, felt like an eternity. Suddenly, I hear the howling wind pick up followed by a loud crack of thunder and lightning....
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...“Just because you don’t understand something doesn’t mean it’s nonsense,” - Lemony Snicket. The fiction novel, A Series of Unfortunate Events: The Bad Beginning, was written by Lemony Snicket. We follow the three Baudelaire children, or, as of recently, the Baudelaire orphans. The eldest child is Violet Baudelaire, she is inventive, inquisitive, and protective. Her hair is brown, she has a purple ribbon tied into her hair, and has pleasant facial features. The next child is Klaus Baudelaire, he is intelligent, curious, and confrontational. He sports glasses, is intelligent looking, and has pleasant facial features. The youngest orphan is Sunny Baudelaire, she knows only a few words, likes to gnaw on objects, and is very loud. She...
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