- #1
pierce15
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Simple question:
Are there any series which we don't know whether or not they converge?
Are there any series which we don't know whether or not they converge?
Series convergence vs. divergence
- Thread starter pierce15
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In summary, series convergence and divergence are two important concepts in mathematics that describe the behavior of infinite sequences of numbers. A convergent series is one in which the terms approach a finite limit as the number of terms increases, while a divergent series is one in which the terms do not approach a limit and instead either tend towards infinity or oscillate between different values. The convergence or divergence of a series can be determined using various tests, such as the ratio test or the integral test. These concepts are crucial in many areas of mathematics, including calculus, analysis, and number theory, and have important applications in fields such as physics and engineering.
- #2
AlephZero
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I'ts fairly easy to invent some. For example, ##a_n = 0## if ##2n## is the sum of two primes, otherwise ##a_n = 1##.
First prove the Goldbach conjecture. Checking the convergence or divergence of the series is then trivial
If you don't like that example, think about the consequences of Godel's incompleteness theorems.
First prove the Goldbach conjecture. Checking the convergence or divergence of the series is then trivial
If you don't like that example, think about the consequences of Godel's incompleteness theorems.
- #3
henpen
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http://mathoverflow.net/questions/65858/series-whose-convergence-is-not-known has a few examples (some of which understandable, too!). I'm particularly fond of Qiaochu Yuan's).
- #4
pierce15
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AlephZero said:I'ts fairly easy to invent some. For example, ##a_n = 0## if ##2n## is the sum of two primes, otherwise ##a_n = 1##.
First prove the Goldbach conjecture. Checking the convergence or divergence of the series is then trivial
If you don't like that example, think about the consequences of Godel's incompleteness theorems.
I don't really understand the implications of the incompleteness theorems, could you briefly explain how they relate to this?
Related to Series convergence vs. divergence
1. What is series convergence and divergence?
Series convergence and divergence refer to the behavior of a mathematical series as the number of terms approaches infinity. A convergent series is one in which the sum of all terms approaches a finite value, while a divergent series is one in which the sum of all terms approaches infinity.
2. How can I determine if a series is convergent or divergent?
There are several tests that can be used to determine the convergence or divergence of a series, such as the ratio test, the integral test, and the comparison test. These tests involve evaluating the behavior of the terms in the series and comparing them to known series with known convergence or divergence behavior.
3. What is the importance of understanding series convergence and divergence?
Understanding series convergence and divergence is crucial in many areas of mathematics and science. It allows us to determine the behavior of infinite sums and to make predictions about the behavior of functions. It also has practical applications in areas such as finance and engineering.
4. Can a series be both convergent and divergent?
No, a series cannot be both convergent and divergent. A series can only have one behavior as the number of terms approaches infinity. However, a series can be conditionally convergent, meaning that it is convergent but its behavior may change if the terms are rearranged.
5. How can I use series convergence and divergence in real-world applications?
Series convergence and divergence have many real-world applications, such as in financial modeling, where they can be used to determine the behavior of investments over time. They are also used in engineering to analyze the stability and behavior of systems. Additionally, series convergence and divergence are important in physics and other sciences to model and predict the behavior of physical phenomena.
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