infinite series - translation to russian
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  3. infinite series

infinite series - translation to russian

SUM OF AN (INFINITE) GEOMETRIC PROGRESSION
Infinite geometric series; Geometric sum; Geometric Series
  • Bit fields for encoding a 32-bit floating point number according to IEEE 754 standard.
  • pages=Appendix B}}</ref> in Appendix B.)
  • Complex geometric series (coefficient ''a'' = 1 and common ratio ''r'' = 0.5 e<sup>iω<sub>0</sub>t</sup>) converging to a circle. In the animation, each term of the geometric series is drawn as a vector twice''':''' once at the origin and again within the head-to-tail vector summation that converges to the circle. The circle intersects the real axis at 2 (= 1/(1-1/2) when ''θ'' = 0) and at 2/3 (= 1/(1-(-1/2)) when ''θ'' = 180 degrees).
  • Close-up view of the cumulative sum of functions within the range -1 < ''r'' < -0.5 as the first 11 terms of the geometric series 1 + ''r'' + ''r''<sup>2</sup> + ''r''<sup>3</sup> + ... are added. The geometric series 1 / (1 - ''r'') is the red dashed line.
  • Elements of Geometry, Book IX, Proposition 35. "If there is any multitude whatsoever of continually proportional numbers, and equal to the first is subtracted from the second and the last, then as the excess of the second to the first, so the excess of the last will be to all those before it."
  • 1/2}} = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.
  • The convergence of the geometric series with ''r''=1/2 and ''a''=1/2
  • Converging alternating geometric series with common ratio ''r'' = -1/2 and coefficient ''a'' = 1. (TOP) Alternating positive and negative areas. (MIDDLE) Gaps caused by addition of adjacent areas. (BOTTOM) Gaps filled by broadening and decreasing the heights of the separated trapezoids.
  • pages=188}}</ref> For example, the area of the biggest overlapped (red) triangle is ''bh''/2 = (2)(1)/2 = 1, which is the value of the first term of the geometric series. The area of the second biggest overlapped (green) triangle is ''bh''/2 = (2''r''<sup>1/2</sup>)(''r''<sup>1/2</sup>)/2 = ''r'', which is the value of the second term of the geometric series. Each progressively smaller triangle has its base and height scaled down by another factor of ''r''<sup>1/2</sup>, resulting in a sequence of triangle areas 1, ''r'', ''r''<sup>2</sup>, ''r''<sup>3</sup>, ... which is equal to the sequence of terms in the normalized geometric series. (MIDDLE) In the order from largest to smallest, remove each triangle's overlapped area, which is always a fraction ''r'' of its area, and scale the remaining 1−''r'' of the triangle's non-overlapped area by 1/(1−''r'') so the area of the formerly overlapped triangle, now the area of a non-overlapped trapezoid, remains the same. (BOTTOM) Aggregate the resulting n+1 non-overlapped trapezoids into a single non-overlapped trapezoid and calculate its area. The area of that aggregated trapezoid represents the value of the partial series. That area is equal to the outermost triangle minus the empty triangle tip: ''s''<sub>n</sub>/''a'' = (1−''r''<sup>n+1</sup>) / (1−''r''), which simplifies to ''s''/''a'' = 1/(1−''r'') when n approaches infinity and {{Pipe}}''r''{{Pipe}} < 1.
  • A geometric interpretation for the same case of common ratio ''r''>1. (TOP) Represent the terms of a geometric series as the areas of overlapped similar triangles. (MIDDLE) From the largest to the smallest triangle, remove the overlapped left area portion (1/''r'') from the non-overlapped right area portion (1-1/''r'' = (''r''-1)/''r'') and scale that non-overlapped trapezoid by ''r''/(''r''-1) so its area is the same as the area of the original overlapped triangle. (BOTTOM) Calculate the area of the aggregate trapezoid as the area of the large triangle less the area of the empty small triangle at the large triangle's left tip. The large triangle is the largest overlapped triangle scaled by ''r''/(''r''-1). The empty small triangle started as ''a'' but that area was transformed into a non-overlapped scaled trapezoid leaving an empty left area portion (1/''r''). However, that empty triangle of area ''a''/''r'' must also be scaled by ''r''/(''r''-1) so its slope matches the slope of all the non-overlapped scaled trapezoids. Therefore, S<sub>n</sub> = area of large triangle - area of empty small triangle = ''ar''<sup>n+1</sup>/(''r''-1) - ''a''/(''r''-1) = ''a''(''r''<sup>n+1</sup>-1)/(''r''-1).
  • The first nine terms of geometric series 1 + ''r'' + ''r''<sup>2</sup> + ''r''<sup>3</sup> ... drawn as functions (colored in the order red, green, blue, red, green, blue, ...) within the range {{Pipe}}''r''{{Pipe}} < 1. The closed form geometric series 1 / (1 - ''r'') is the black dashed line.
  • Another geometric series (coefficient ''a'' = 4/9 and common ratio ''r'' = 1/9) shown as areas of purple squares. The total purple area is S = ''a'' / (1 - ''r'') = (4/9) / (1 - (1/9)) = 1/2, which can be confirmed by observing that the [[unit square]] is partitioned into an infinite number of L-shaped areas each with four purple squares and four yellow squares, which is half purple.
  • The convergence of the geometric series with ''r''=1/2 and ''a''=1
  • A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2.
  • A compact illustration of different perspectives on the geometric series to make perspective switching easier. Quadrant A1 contains the '''why''' perspective briefly describing why geometric series matter. Quadrant A2 contains the '''how''' perspective listing a snippet of Julia code that computes the geometric series. Quadrant A3 contains the '''what''' (algebra) perspective with the algebraic proof of the geometric series closed form formula. Quadrant A4 contains the '''what''' (geometry) perspective showing three steps of the geometric proof of the geometric series closed form formula. And the right margin shows the timeline of some of the historical insights about the geometric series.

infinite series      
бесконечный ряд
infinite series      

математика

бесконечный ряд

infinite series      
бесконечный ряд

Definition

дрифт
муж., мор. конец борта при окончании шканец, юта и бака.
| Разность между толщиной болта и размером дыры или гнезда его: простор, зазор.
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Wikipedia

Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series

1 2 + 1 4 + 1 8 + 1 16 + ⋯ {\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots }

is geometric, because each successive term can be obtained by multiplying the previous term by 1 / 2 {\displaystyle 1/2} . In general, a geometric series is written as a + a r + a r 2 + a r 3 + . . . {\displaystyle a+ar+ar^{2}+ar^{3}+...} , where a {\displaystyle a} is the coefficient of each term and r {\displaystyle r} is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential.

The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series terms (without any of the additions) is called a geometric sequence or geometric progression.

https://en.wikipedia.org/wiki/Geometric series
What is the Russian for infinite series? Translation of &#39infinite series&#39 to Russian
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