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Some of the amounts of rows with squares of integrals
Unusual next faced the author of this paper in studying the phenomena related to the Lorentz factor. Under certain conditions the sum of this infinite series will always have the same value. First you need to introduce the function \[h(t) = a\,t \qquad (1)\] which will be used in subsequent expressions. It: \(a\) is some constant, \(t\) — variable on which the integration is performed. Imagine the following sum of the series: \[a^2 \sum \limits_{n=0}^{\infty} \left[ \int \limits_0^{1/a} h(t)^n \sqrt{1 - h(t)^2} \,\Bbb{d} t \right]^2 = 0.816 \qquad (2)\] the Sum of this series is always equal to \(0.816\), any \(a\). The following amount will change the upper limit, but it still will be the same: \[a^2 \sum \limits_{n=0}^{\infty} \left[ \int \limits_0^{0.5/a} h(t)^n \sqrt{1 - h(t)^2} \,\Bbb{d} t \right]^2 = 0.244 \qquad (3)\] Same and the following sum: \[a^2 \sum \limits_{n=1}^{\infty} \left[ \int \limits_0^{1/a} h(t)^n \sqrt{1 - h(t)^2} \,\Bbb{d} t \right]^2 = 0.2 \qquad (4)\] Here is a summation of a number does not begin with zero, and with unit. This amount also will always equal, for any \(a\). Changing the upper limit of integration and the initial value of \(n\) is possible to obtain different values of stable sums of a series.