Understanding mathematics needs more imagination than an artist has. Some concepts look simple, but when we got the insight it isn't as simple as it was. For example consider a simple infinite series
1-1+1-1+1-1+...........∞
If I ask what would be the sum of this series, anyone can say it is zero, because,
(1-1)+(1-1)+(1-1)+(1-1)+......∞
But, if we think other possibility ,like this,
1+(-1+1)+(-1+1)+(-1+1)+......∞
then we will get sum as one.
This series is called Grandi's series. In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, …, which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent.
What is the problem here?. Why it is giving two different accumulation points?. The answer is simple. This problem is not just a problem of summing a series. There is another fundamental question hidden behind this. The questions is "Is infinity even or odd?" . Even infinity leads to zero and odd infinity leads two one.
Answer to this question is not easy. I don't think there exists a correct answer to this question. How ever what I feel is we may consider both answers are correct. This is similar to the Hilbert's first problem "Continuum hypothesis ", one of the first proved example of Gödel's incompleteness theorem.Paul Cohen proved that we can assume "continuum hypothesis" either true or false without violating other mathematical axioms. At last it is proved that there is no answer to a question Hilbert most wanted. Even Paul Erdös was so eager to know the answer. Once he was asked "what would you ask if God appears in front of you?", he immediately reapplied that he would ask "whether Continuum hypothesis is true or false?" and he continued that "God would probably say three answers , first answer would be Gödel and Cohen already thought you, Second , you don't have enough intelligences to understand the proof and the the third, Still I am searching for the proof from the time of creation, and he finished by commenting that the third answer would be more appropriate.
This series is called Grandi's series. In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, …, which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent.
What is the problem here?. Why it is giving two different accumulation points?. The answer is simple. This problem is not just a problem of summing a series. There is another fundamental question hidden behind this. The questions is "Is infinity even or odd?" . Even infinity leads to zero and odd infinity leads two one.
Answer to this question is not easy. I don't think there exists a correct answer to this question. How ever what I feel is we may consider both answers are correct. This is similar to the Hilbert's first problem "Continuum hypothesis ", one of the first proved example of Gödel's incompleteness theorem.Paul Cohen proved that we can assume "continuum hypothesis" either true or false without violating other mathematical axioms. At last it is proved that there is no answer to a question Hilbert most wanted. Even Paul Erdös was so eager to know the answer. Once he was asked "what would you ask if God appears in front of you?", he immediately reapplied that he would ask "whether Continuum hypothesis is true or false?" and he continued that "God would probably say three answers , first answer would be Gödel and Cohen already thought you, Second , you don't have enough intelligences to understand the proof and the the third, Still I am searching for the proof from the time of creation, and he finished by commenting that the third answer would be more appropriate.
1 comment:
hehehe.. you know this is a divergent series!!!
not a converging one, just like ur way of treating it!
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