In Deriving the Extraordinary Euler Sum , we derived one of Euler's most celebrated results: $latex \displaystyle 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^3} + ... = \frac{\pi^2}{6}\quad\quad\quad(1)$ Now, we aim to provide a rigorous proof of this statement. First, we expressed the partial sum of the left-hand side as follows: $latex \displaystyle \sum\limits_{i=1}\frac{1}{i^2} = \sum\limits_{i=1}\frac{1}{(2i)^2}…

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06.04.2024

Harnessing Feynman’s Integral Technique |

Show that $latex \displaystyle\int_{-\infty}^{\infty}e^{-x^2}\;dx = \sqrt{\pi}$ This integral is renowned in mathematics as the Gaussian integral. Its evaluation poses a challenge due to the absence of an elementary antiderivative expressed in standard functions. Conventionally, one method involves "squaring the integral" and subsequently interpreting the resulting double integral in polar coordinates. However, an alternative approach, which…

Vroom

06.04.2024

Sandwich Theorem for Functions 2 |

For given functions $latex f(x), g(x),$ and $latex h(x),$ $latex \left(\forall x : f(x) \le g(x) \le h(x), \lim\limits_{x \rightarrow \infty} f(x) = \lim\limits_{x \rightarrow \infty} h(x)=L\right) \implies \lim\limits_{x \rightarrow \infty} g(x) = L.$ The given condition gives $latex \forall \epsilon > 0, \exists x_1^* \ni x>x_1^*, |f(x)-L|<\epsilon$ and $latex \forall \epsilon > 0, \exists…

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06.04.2024

Introducing Feynman’s Integral Method (MAA Meeting, Spring 2024) |

Introducing Feynman's Integral Method Michael Xue Indiana Section MAA Spring 2024 MEETINGMarian University - IndianapolisApril 5-6, 2024 "The first principle is that you must not fool yourself and you are the easiest person to fool." -- Richard Feynman Feynman's Integral Method Leibniz : "$latex \displaystyle \frac{d}{d\beta}\int\limits_{a}^{b}f(x, \beta) \;dx = \int\limits_{a}^{b}\frac{\partial}{\partial\beta}f(x, \beta)\;dx$'' Feynman: "To evaluate hard…

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Science

An Analytic Proof of the Extraordinary Euler Sum |

Harnessing Feynman’s Integral Technique |

Sandwich Theorem for Functions 2 |

Introducing Feynman’s Integral Method (MAA Meeting, Spring 2024) |

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