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