Cesaro summation Essays - Mathematical Analysis, Mathematics

Cesaro summation Essays - Mathematical Analysis, Mathematics Cesaro summation Does 1 - 1 + 1 - 1 = 1/2? Submitted to: Mr. Mark Submitted by: Putri Introduction In mathematical analysis, which includes the study sequences and series, the infinite series 1 - 1 + 1 - 1 + ., can be also called Grandi's series. This is named after an Italian mathematician, philosopher and a priest, Guido Grandi. It is known as a divergent series which means that it lacks a sum in the conventional sense. Contrarily, the Cesaro sum is . Cesaro summation allocates values to some infinite sums which are not convergent usually. This sum is defined as the limit of the arithmetic mean of the partial sums of the series. It is named after Ernesto Cesaro, an Italian analyst (1859 - 1906). Most individuals often acknowledge statements and proofs regarding the Cesaro summation as an implication of Eilenberg-Mazur swindle. One example can be drawn from the Grandi's series. It is usually applied to the Grandi's series and the conclusion is that the sum of that series is . This result is totally be disproven. What is the sum of the infinite sequence 1, -1, 1, -1, 1? From my opinion, the two major intuitive answers are either that it sums to zero or no sum at all. If we arrange the pattern into pairs, then each pair (1, -1), which in result gives a 0. However if we arrange the pattern by first leaving the 1, then grouping pairs of (-1,1) would end up giving a sum of 1. First of all, it's worth seeing why we shouldn't just use our formula for an infinite geometric series: In the formula, r is the multiplicative constant of -1. The infinite geometric formula requires that the absolute value of r is less than 1, or else the series will not converge. The Cesaro Method Let a n } be a sequence, and let be the k th partial sum of the series. The series is called Cesaro summable, with Cesaro sum , if the average value of its partial sums tends to A : Simply put, the Cesaro sum of an infinite series is the limit of the arithmetic mean of the first n partial sums of the series, as n approaches infinity. If a series is convergent, then it can be summed and called Cesaro summable and its Cesaro sum is 0. For any convergent sequence, the corresponding series will be Cesaro summable and the limit of the sequence occur simultaneously with the Cesaro sum. Mathematical Computations Using the Cesaro method, If a n = (1) n +1 for n 1. It means, a n } is the sequence. 1,-1,1,-1... Then the sequence of partial sums s n } is 1, 0, 1, 0, ... so while the series not converge, if we calculate the terms of the sequence ( s 1 + + s n ) / n } we get: so that So, by using various different methods, I have shown that this series "should" have a summation of 0 (grouping in pairs), or that it "should" have a sum of 1 (grouping in pairs after the first 1), or that it "should" have no sum as it simply oscillates, or that it "should" have a Cesaro sum of , unsurprisingly it caused so much dismay amongst mathematicians. Conclusion The Grandi's series got its name after a few hundred years of mathematical debate to find what the correct summation was. This is quite a long time. And one of the solution to it was the finding of Cesaro summation which gives a result of . To sum up, we can never know the Cesaro sum if no one discover or study the series in depth, so we can't just guess the real answer based on our intuition, it could be correct but in reality, there is so much more that needs to be discovered. The series can actually be extended to a more complex one, , this includes the exploration of the sum of powers of i. The Ce saro sum shows how different proofs can sometimes lead to different and unforeseen results. What does this statements say about the nature of proof? Reflection

5 Essential Tips for a KILLER LinkedIn Summary

5 Essential Tips for a KILLER LinkedIn Summary Many of you might have heard that LinkedIn profiles are about to undergo another facelift. Rumor amongst career professionals has is that the new profile will highlight the Summary section in a new way- making it more important than ever to have a compelling Summary to offer your readers. One of the most popular services The Essay Expert offers is writing the LinkedIn Profile Summary. We love interviewing people and writing copy for them; but you can also do it yourself if you are a good writer and know the message you want to get across. If you are a do-it-yourselfer, some tips for writing a great LinkedIn Summary might come in handy! Well, you’re in luck. CareerCast published one of my articles that provides five tips for a KILLER LinkedIn Summary; and although I agreed not to post the entire article on my own blog, I wanted to share the link with you. Here it is! 5 Essential Tips for a KILLER LinkedIn Summary As a preview, the tips are: Never leave the Summary section blank! Use all 2,000 characters. Tell stories. Break it up (graphically). Include your contact information [NOTE: I no longer suggest doing this unless you have space to spare, since the Contact Information section has recently been made more prominent.] I will likely write a sequel to this article soon, so please take a look at the first one and stay tuned for more useful advice! For samples of The Essay Expert’s LinkedIn Summary statements, see our LinkedIn Profile Samples. And if you are looking for a professional LinkedIn Profile Writing service, please visit our LinkedIn Profile Writing page to learn more about our services and rates. Do you have more ideas about how to write a KILLER LinkedIn Summary? Please share below†¦ you might even get quoted in my next article!

Amendments Essay Example | Topics and Well Written Essays - 1250 words

Amendments - Essay Example Prior to these amendments, some states had already made their own ratifications of the constitutional contingents, which prompted the need to limit the powers of the federal government with regard to investigation, prosecution and punishment of crime (Calvin, 2009). The original constitution offered very little provisions touching on criminal processes (Calvin, 2009). A majority of the constitutional amendments were mainly focused on addressing issues relating to criminal processes. Fifteen of the twenty six rights contained in the first eight amendments to the constitution focus mainly on addressing the criminal and judicial processes. These amendments were created based on a perspective that emerged in the late 18th century about individual rights. Initially, the amendments were meant to only be applicable to the federal government, but with time, states started adopting most of the provisions in the amendments (David & James, 2008). In order to fully understand the impact of these amendments on the judicial processes in both juvenile and adult courts, it is important to first and foremost understand the phrase â€Å"principles of due process† as used in the 5th amendment (John, 2006). Due process of law implies that all laws be applied equally and fairly to all American citizens, and most importantly those suspected to have committed crimes. The fourth amendment protects citizens from unreasonable seizures and searches of their houses, and their property (John, 2006). This amendment limits the powers of the central government when conducting criminal investigations. The government cannot unlawfully enter a suspect’s house or any of his or her property, papers among other things to look for evidence. Search warranties supported by affirmation or Oath have to be issued so as to enable the police to conduct searches. Interpretations of the amendment by the Supreme Court restricted police searches and seizures only to the suspect’s bodies an d clothes, the things they have in plain sight, the places or things that the accused is in control or which he could reach and property where the police strongly suspect that a person could be in danger (Calvin, 2009). This amendment had a great impact on criminal proceedings. It restricted the powers of the government in investigating crimes (Calvin, 2009). The amendment made it unlawful for the federal police to break in to suspect’s houses with the intention of searching and seizing some of the property purported to provide evidence of a crime. The police need to follow certain procedures, including obtaining a search warrants so as to be able to conduct houses and properties belonging to criminal suspects. The Fifth Amendment protects individuals from ‘double jeopardy’ (Calvin, 2009). This refers to the process of initiating a second trial for a person suspected to have committed the same crime. This amendment requires that before a person is any serious cri minal offence, there must be a thorough investigation by the jury linking him to the offense. This amendment also gives the defendant in a court process the right not to testify against himself, commonly known as the right to remain silent. The main intention of the