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Tag Archives: infinite_series
Fun With Math: How To Make A Divergent Infinite Series Converge by Kevin Knudson
“Fun With Math: How To Make A Divergent Infinite Series Converge” is an interesting article about how to make the harmonic series converge by systematically discarding terms. Shockingly, the article appeared in Forbes. ‘I was having dinner with a visiting … Continue reading
Posted in Mathematics
Tagged harmonic_series, infinite_series
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An Interview with Neil Sloane the Creator of the Online Encyclopedia of Integer Sequences
‘Neil Sloane is considered by some to be one of the most influential mathematicians of our time. That’s not because of any particular theorem the 75yearold Welsh native has proved, though over the course of a more than 40year research … Continue reading
Posted in Mathematics
Tagged infinite_series, neil_sloane
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Proof of \( e^{x}e^{y} = e^{x+y} \) Using Infinite Series
The easiest method of proving \( e^{x}e^{y} = e^{x+y} \) is to use logarithms. Indeed, this is the method most often used in introductory analysis texts. However, if one develops the theory of exponential functions before that of logarithms, this … Continue reading
Posted in Mathematics
Tagged cauchy_product, exponential_functions, infinite_series
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Cauchy Condensation Test
Determining the convergence or divergence of infinite series is much like calculating integrals. There are some rules that can be readily applied in simple cases, while for more difficult problems, a combination of rules, intuition, and trial and error are … Continue reading
Posted in Mathematics
Tagged augustin_cauchy, cauchy_condensation_test, infinite_series
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Proof of the Week for 20121001
Posted in Mathematics
Tagged infinite_series, proof_of_the_week
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Proof of the Week for 20120917
Posted in Mathematics
Tagged infinite_series, karl_weierstrass, mtest, proof_of_the_week
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Proof of the Week for 20120813
Cauchy convergence criterion for a series of real numbers: The series \(\sum_{n=1}^{\infty} a_{n}\) converges if and only if for \(\epsilon \gt 0\), there exists \(N \in \mathbb{N}\) such that whenever \(n, m \ge N\), \(a_{m+1} + a_{m+2} + \cdots + … Continue reading
Posted in Mathematics
Tagged augustin_cauchy, calculus, infinite_series, proof_of_the_week
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