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

20 Words Meaning Being or Existing in the Past

20 Words Meaning Being or Existing in the Past 20 Words Meaning Being or Existing in the Past 20 Words Meaning Being or Existing in the Past By Mark Nichol This list features former and eighteen other adjectives (and a prefix) that can be used to refer to a position no longer held or a state no longer active, exclusive of the synonyms for original. 1. antecedent: Though this word is usually seen in noun form, it can also be an adjective, as in â€Å"She wants to answer the antecedent question.† 2. earlier: This comparative of early, like most words on this list, can function as an adjective as well as an adverb; in the former usage, one can say, for example, â€Å"The earlier iteration had many flaws.† 3. erstwhile: This somewhat stuffy, self-conscious synonym for former was originally an adverb, as in â€Å"The two were erstwhile adversarial† but has been in use as an adjective since the early 1900s. The first syllable from this word from Old English is from the superlative for ere (â€Å"before†). 4. ex-: This prefix is attached, nearly always with a hyphen, to a noun to signify that a status is no longer current, as in â€Å"All four living ex-presidents attended the event.† (The exception is when an en dash is used in place of the hyphen to signal that what follows the prefix is an open compound, as in â€Å"All four living ex–vice presidents attended the event.† It might be better, however, just to use former or another alternative.) The prefix, which originally meant â€Å"from† or â€Å"out of† but now has several senses, is also sometimes used as a stand-alone word, especially in reference to a former significant other, as in â€Å"Oh, I saw your ex the other day.† 5. former: This is the default word to describe the sense of something no longer being what it was; it can also be used as the antonym of latter to refer to the first of two things. 6. late: When used on its own to modify a person’s name (â€Å"the late John Smith†), it means that the person is deceased. â€Å"Late of,† however, is an outdated way of saying that someone is no longer associated with something, as in â€Å"Mary Jones, late of Centerville† (meaning that she recently lived there). 7. onetime: This synonym of former (â€Å"Mary Jones is a onetime resident of Centerville†) may also mean â€Å"done or happening one time only†). The word is sometimes hyphenated when the latter meaning is intended. 8. old: Old distinguishes something from something else from an earlier time that was replaced, as in â€Å"I think that the old design looks better.† 9. original: This word can be used in place of former or a synonym to refer to an initial state, as in â€Å"The original plans called for a large entrance.† 10. other: This word occasionally substitutes for former, as in â€Å"In other times, we might not have given it a second thought†; a similar usage is, for example, â€Å"the other day,† referring to a recent day. 11. past: This word is used narrowly in the sense of someone who no longer holds a position, as in â€Å"past president.† 12. precedent: Like the similar word antecedent, this term is usually used as a noun but can function as an adjective, as in â€Å"A precedent event in her life turned out to be a character-defining one.† 13. preceding: Preceding has the same sense as precedent and is used more commonly. 14. preexisting: This word means â€Å"existing in an earlier time.† (Note that the prefix pre- is not attached to the root word with a hyphen.) 15. previous: Previous can mean â€Å"existing before in general† or â€Å"existing immediately before,† depending on whether one writes, for example, â€Å"a previous administration† or â€Å"the previous administration.† 16. prior: This word is identical in meaning to previous, though it is less likely than that word to be used to refer to mean â€Å"immediately previous†; it also has the sense, rarely employed, of â€Å"being more important because it came first.† 17. quondam: This direct borrowing from Latin was originally an adverb and a noun, but those usages are obsolete, and adjectival use is rare and often considered overly formal. 18. sometime: Sometime began as an adverb. Later, it acquired the sense of â€Å"at some future time,† as in â€Å"I’ll see you again sometime,† and finally developed an adjectival sense. 19. then: The adverbial use of this word (â€Å"I’ll go, then†) and its function as a conjunction (â€Å"First, I waved, and then I called out to her†) were followed by the development as an adjective meaning â€Å"being at that time†; it should be used in place of former or most other synonyms to indicate that the state existed during the time being discussed, as in â€Å"In 1968, then California governor Ronald Reagan considered running for president.† (Note that then is not hyphenated to the following word.) 20. whilom: This synonym for former (and formerly) is archaic and rarely used anymore. Want to improve your English in five minutes a day? Get a subscription and start receiving our writing tips and exercises daily! Keep learning! 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